In the Catholic tradition, several "apocryphal" books are used in addition to the same Old Testament canon used by Protestants.
The books known as "apocryphal" to Protestants are defined by Catholics as "Deuterocanonical" (a second canon). They come from the Septuagint, a Greek translation (with these additional books) of the Hebrew Tanakh. Later, around the 4th century, the Old Testament was translated by St. Jerome into Latin as part of the Vulgate. Besides the Catholic Church, the Constantinople-based Eastern Orthodox Church also received Deuterocanonical books as canonical.
Catholic Church considers the Deuterocanonicals on the same level as other books in the OT and NT canon. It is not a separate list, as expressed in the Catechism of the Catholic Church (CCC) article 120. The Deuterocanonicals are also "infallible", as expressed in CCC article 107; they are without error.
This topic has been debated from the beginning, even by St. Jerome himself as he made the Vulgate. He considered some of the books and texts not found in Hebrew to be apocryphal, as expressed in his prologue to the book of Kings, Esdras, Solomon, Jeremias and Judith. But later, throughout the Middle Ages, the Vulgate was accepted as the official translation by the Church (Western Catholic and Eastern Orthodox). After Luther's Germanic Bible in 1534, this issue became one of the points of difference between the "Old Churches" (Western Catholic and Eastern Orthodox) and "Protestant and Reformed Churches" (Luther and Calvin).
Parts of this answer is taken directly from the Catechism of the Catholic Church (1994) which carries the seal of Imprimi Potest, by which the Catholic Church recognizes the publication to be free of doctrinal error (as I understand it).
On the Inspiration and Truth of Sacred Scripture, attesting to divine authorship:
105 God is the author of Sacred Scripture. "The divinely revealed realities, which are contained and presented in the text of Sacred Scripture, have been written down under the inspiration of the Holy Spirit."
And attesting to inerrancy of all, including the deuterocanonical books (emphasis mine):
107 The inspired books teach the truth. "Since therefore all that the inspired authors or sacred writers affirm should be regarded as affirmed by the Holy Spirit, we must acknowledge that the books of Scripture firmly, faithfully, and without error teach that truth which God, for the sake of our salvation, wished to see confided to the Sacred Scriptures."
Therefore, the books in question are viewed as inerrant and on the same level as the rest of the Biblical canon, old and new testaments.
To speak to Irawan's point as to how Catholics view scripture, the relationship between scripture and "tradition" is as follows (CCC 80 ff):
One common source. . .
80 "Sacred Tradition and Sacred Scripture, then, are bound closely together, and communicate one with the other. For both of them, flowing out from the same divine well-spring, come together in some fashion to form one thing, and move towards the same goal."40 Each of them makes present and fruitful in the Church the mystery of Christ, who promised to remain with his own "always, to the close of the age".
. . . two distinct modes of transmission
81 "Sacred Scripture is the speech of God as it is put down in writing under the breath of the Holy Spirit."
"and [Holy] Tradition transmits in its entirety the Word of God which has been entrusted to the apostles by Christ the Lord and the Holy Spirit. It transmits it to the successors of the apostles so that, enlightened by the Spirit of truth, they may faithfully preserve, expound and spread it abroad by their preaching."
82 As a result the Church, to whom the transmission and interpretation of Revelation is entrusted, "does not derive her certainty about all revealed truths from the holy Scriptures alone. Both Scripture and Tradition must be accepted and honoured with equal sentiments of devotion and reverence."
Apostolic Tradition and ecclesial traditions
83 The Tradition here in question comes from the apostles and hands on what they received from Jesus' teaching and example and what they learned from the Holy Spirit. the first generation of Christians did not yet have a written New Testament, and the New Testament itself demonstrates the process of living Tradition.
Tradition is to be distinguished from the various theological, disciplinary, liturgical or devotional traditions, born in the local churches over time. These are the particular forms, adapted to different places and times, in which the great Tradition is expressed. In the light of Tradition, these traditions can be retained, modified or even abandoned under the guidance of the Church's Magisterium.
The origins of the Catholic OT canon is rather a lengthy discussion, which can be found here. The gist of it is (emphasis mine):
It now seems more probable that at the time of Christianity's birth, closed collections of the Law and the Prophets existed in a textual form substantially identical with the Old Testament. The collection of “Writings”, on the other hand, was not as well defined either in Palestine or in the Jewish diaspora, with regard to the number of books and their textual form. Towards the end of the first century A.D., it seems that 22 books were generally accepted by Jews as sacred, but it is only much later that the list became exclusive. When the limits of the Hebrew canon were fixed, the deuterocanonical books were not included.
...
In the West, the use of a larger collection of sacred books was common and was defended by Augustine. When it came to selecting books to be included in the canon, Augustine (354-430) based his judgement on the constant practice of the Church. At the beginning of the fifth century, councils adopted his position in drawing up the Old Testament canon. Although these councils were regional, the unanimity expressed in their lists represents Church usage in the West.
From this information, assuming it's in fact accurate, it could be argued that the reformers rejected an OT canon which the Christian Church had used for well over a millennia; I have read arguments by Catholic writers which suggest that the Jewish canon was altered (or perhaps finalized) well after the Christian church was established, but I cannot find references to support that.
The books that non-Catholics (generally Protestants) call the Apocrypha are called by Catholics the Deuterocanonical books (from the Greek δεύτερος, second; and κανῶν, literally a straight rod or bar, hence a unit of measure, or, by extension, a list).
They are called the “second” canon because, when the canon of Scripture was being debated in the first centuries of the Church, there was some debate as to whether they should be included in the canon or not.
The Old-Testament deuterocanonical books include Tobit, Judith, Wisdom (or Wisdom of Solomon), Sirach (or Ecclesiasticus), Baruch (including the so-called Letter of Jeremiah), 1 and 2 Maccabees, and some additional sections in Esther and Daniel. (See the Wikipedia article on this subject.)
Note that there is also a New-Testament deuterocanon; there was also considerable debate as to whether Hebrews, James, 2 Peter, 2 John, 3 John, Jude, and Revelation should be included in the canon. (The New-Testament deuterocanon should not, however, be confused with truly apocryphal books from the New-Testament era, such as Protoevangelium of James, the Pastor of Hermas, and especially from much later Gnostic texts such as the Gospel of Thomas.)
The Canon as the Catholic Church recognizes it seems to have stabilized around the end of the fourth century. For example, in 382, a synod in Rome issued the following decree, reaffirmed by Pope Damasus I:
Now the Divine Scriptures must be dealt with, regarding which books the universal Catholic Church receives, and which must be avoided. The order of the Old Testament begins: Genesis (one book); Exodus (one book); Leviticus (one book); Numbers (one book); Deuteronomy (one book); Joshua (one book); Judges (one book); Ruth (one book); Kings (four books) [equivalent to 1-2 Samuel and 1-2 Kings in English Bibles]; Paralipomenon (two books) [1-2 Chronicles in English]; 150 Psalms (one book); Solomon (three books: Proverbs, one book; Ecclesiastes, one book; Song of Songs, one book); likewise Wisdom (one book), Ecclesiasticus [Sirach] (one book). Likewise the order of the Prophets: Isaiah (one book); Jeremiah (one book) with Cinoth (that is, his Lamentations); Ezekiel (one book); Daniel (one book); Hosea (one book); Amos (one book); Micah (one book); Joel (one book); Obadiah (one book); Jonah (one book); Nahum (one book); Habakkuk (one book); Zephaniah (one book); Haggai (one book); Zechariah (one book); Malachi (one book). Likewise, the order of the historians: Job (one book); Tobit (one book); Ezra (two books); Esther (one book); Judith (one book); Maccabees (two books) (my translation, highlighting places where deuterocanonical books are mentioned; see Denzinger-Hünermann [DH] 179).
(The list continues in No. 180 for the New Testament, including the canon we are familiar with. Curiously, the book of Baruch is not mentioned, but it was included in St. Jerome’s translation of the Bible—commissioned by Pope Damasus. It is possible that the decree is joining Baruch together with another book—probably Jeremiah. It should be noted that Jerome was not at first in favor of the canonicity of the Deuterocanonicals—but he accepted the judgment of the Church.)
A local council in Carthage (at which St. Augustine of Hippo was doubtless present) published a similar list in 397:
These are the canonical Scritpures: Genesis, Exodus, Leviticus, Numbers, Deuteronomy, Joshua, Judges, Ruth, 1-4 Kings, 1-2 Paralipomenon, Job, the Davidic Psalter, five books of Solomon [which would be Proverbs, the Song of Songs, Ecclesiastes, Sirach, and Wisdom], twelves books of Prophets [the so-called minor prophets: Hosea, Joel, Amos, Obadiah, Jonah, Michah, Nahum, Habakkuk, Zephaniah, Haggai, Zechariah, and Malachi], Isaiah, Jeremia [and Lamentations], Daniel, Ezechiel, Tobit, Judith, Esther, 1-2 Ezra, 1-2 Maccabees. Of the New Testament: four Gospels, the Acts of the Apostles (one book), thirteen epistles from Paul the Apostle, one epistle from him to the Hebrews, two from Peter, three from John, one from James, on from Jude, and the Apocalypse of John (DH 186; again, the book of Baruch seems to be included together with Jeremiah, since in the Vulgate it comes right after Lamentations and finishes with a Letter from Jeremiah).
As can be seen, these synods made no particular distinction in their lists between protocanonical (the books that practically all Christians agreed were canonical) and deuterocanonical (debated) works.
In the West, the issue of the Canon of Scripture remained fundamentally uncontroversial until attempts were made in the Council of Florence (1438-1445) to return to full unity with the Eastern Orthodox and other Eastern Churches not then in union with Rome. It turned out that the Eastern Churches had an even broader canon, including books such as 3 Maccabees and 1 (or 3) Esdras. The Council of Florence declared that only those books recognized in the West since the synods of Rome and Carthage were canonical. (See the bull of union with the Coptic Church, Cantate Domino, DH 1335, which this time includes the book of Baruch explicitly).
Finally, in the aftermath of the Protestant Reformation, the Council of Trent dogmatically affirmed the previous decrees, thus permanently closing the canon. After listing the same books as in the previous lists, the council says
If someone, however, were not to accept these books, in their entirety and with all their parts, as they have customarily been read in the Catholic Church and as they are found in the old Latin Vulgate edition, as sacred and canonical, and knowingly and deliberately were to have contempt for the traditions, let him be anathema [the traditional formula for solemnly defining a dogma] (DH 1504, my translation).
As regards the origins of the Old-Testament Deuterocanonicals, they are generally works written after the Exile (587-539 B.C.), and many probably after Alexander the Great’s conquest of the Middle East. Some of the books appear to the translations from Hebrew—Baruch, Tobit, Sirach, and Judith—and the others appear to have been written directly in Greek (including, apparently, the additions to Daniel and Esther). All of these were included in the Septuagint translation of the Old Testament, prepared in Alexandria in the third century B.C. (although apparently not all of the manuscript traditions contain all of the Deuterocanonical works accepted today.)
Inclusion in the Septuagint was not the decisive condition for inclusion in the Old-Testament canon, and neither was being cited in the New Testament; rather, the definitive criterion was continuous use in the Sacred Liturgy, especially the Celebration of the Eucharist. (It should be noted at this point that most of the Jews in the second century specifically excluded the Deuterocanonical books from their Tanakh, or Bible; this exclusion was, however, largely because the Jews at the time had become somewhat suspicious of works that did not have an extant Hebrew original.)
The Deuterocanonicals differ from the Protocanonicals only in that they lack an extant Hebrew original, and above all because there was considerable debate in the early Church as to whether they should be included in the Canon.
It should be specified that the Church considers the Scriptures not merely infallible but inerrant and inspired. As Vatican II’s Dei Verbum puts it,
In composing the sacred books, God chose men, and while employed by Him they made use of their powers and abilities, so that with Him acting in them and through them, they, as true authors, consigned to writing everything and only those things which He wanted. Therefore, since everything asserted by the inspired authors or sacred writers must be held to be asserted by the Holy Spirit, it follows that the books of Scripture must be acknowledged as teaching solidly, faithfully and without error that truth which God wanted put into sacred writings for the sake of salvation (no. 11).
(The original Latin text makes it clear the the Council Fathers intend to teach that inerrancy applies to the entire Bible—that is, that all of the truths found therein are necessary for our salvation.)
Since the Deuterocanonicals are included in the Canon, it follows that they—like all the other books—are inspired and inerrant.
It does not follow, however, every book in the Bible has the same value. Clearly, the four canonical Gospels are far more valuable than (say) the book of Jude. Analogously, the books of the Pentateuch in the Old Testament more fundamental than, say, the Book of Judith. The relative importance of a book does not, however, take way from its inspired character.
The Old-Testament Deuterocanonicals have an importance that depends on their contents (as with all the other books of the Bible). The books of the Maccabees have an importance similar to that of the other historical books. The book of Baruch has value as a prophetic work. Wisdom (a fascinating fusion of Jewish wisdom with Platonic philosophy) and Sirach have an importance similar to the other Wisdom literature.
The additions to Daniel and Esther, as well as Tobit and Judith, do not seem to be historical in the strict sense. It seems that they were written long after the fact, and Judith in particular has details that make it historically improbable (e.g., Nebuchadnezzar was king of the Chaldeans, not the Assyrians). That does not mean that thy fail to be inspired or inerrant; it was simply not the intention of their authors to make historically accurate accounts. Judith seems to be a kind of “historical fiction” in our modern parlance.
All of these books continue to be used in the Catholic Church’s liturgy, both in the Mass and in the Liturgy of the Hours.
In the Catholic tradition, there 3 types of historical writings dealt with here: biblical writings and/or non-biblical writings known as Apocrypha.
The OP states that there are several books "used in addition to the same Old Testament canon used by Protestants", but also calls these books Apocrapha which betrays the OP's Protestant tradition since the Catholic tradition regards these works as deuterocanonical (or second-canon, with deutero meaning "second").
These works appear to have been formally categorized at the Council of Rome in 382 AD by Pope Damasus I via the Decretum Gelasianum, or Gelasian Decree. This decree formally approved the Damasine List - the listing of the proposed canonical books of the bible and several apocrypha. The decree occurs in 5 sections with Section 2 listing the canonical works and section 5 listing the apocryphal works.
In addition to the 66 books of the Protestant Bible, the Damasine List lists the following additional works of the Old Testament:
This makes for a total of 73 books in the Catholic bible. In addition the Damasine List lists the following apocryphal works:
This list totals 62 Apocrapha. A full English translation of the Decretum Gelasianum may be found here.
The Deuterocanonical works come from the Septuagint - the Greek translation of the Old Testament. During the times of and shortly after Jesus, this translation of the Torah was often used by Jewish peoples in the practicing of their faith. Jews in Rome quickly found however that issues with translation and related misunderstandings arose from this text and for this and various other reasons eventually returned to using texts in Hebrew exclusively (and thus Jewish peoples now also regard these books as apocryphal) Furthermore, during the translation process additional works were added to the Septuagint. By comparing the Masoritic Text and the Dead Sea scrolls, we can see that the the Deuterocanonical works were added to or included with the Septuagint sometime between the 3rd century BCE and 132BCE. What is interesting about these additions is that they did not include other well-known apocrypha like Enoch and Jubilees, which indicates their earlier rejection by the Jewish community. An excellent discussion of this in more detail may be found here.
In addition to the books listed above, the Septuagint also included Additions to Daniel and Esther:
Because the Masoritic text did not include the Deuterocanonical works, it was believed by Martin Luther after the Protestant Reformation that these works should not be considered canonical. This version of the canon is known as "Luther's Canon" and is the basis for the Protestant's version of the Bible. Luther also questioned the inclusion of the books of Hebrews, James, Jude and Revelation, though he didn't formally exclude them from his canon, but did relegate them to secondary works and some Bibles still place these books last as a result.
The final, formal decision to standardize the production of Protestant Bibles to conform to the Luther Canon was made in 1825 by the British and Foreign Bible Society.
What is also interesting is that the Psalms of Solomon, 3 Maccabees, 4 Maccabees, the Epistle of Jeremiah the Book of Odes, the Prayer of Manasseh and Psalm 151 are included in some copies of the Septuagint, yet these were not included in the Damasine List. It is possible that the author of said list simply did not have access to a copy of the Septuagint with those books when forming his list.
As these works are considered canon by the catholic church they hold the same status as all other works in the Protestant Bible according to the Catholic church. This was reaffirmed at the Council of Trent at the fourth session wherein it was declared that all works, canonical and dueterocanonical
having been dictated, either by Christ's own word of mouth, or by the Holy Ghost, and preserved in the Catholic Church by a continuous succession. ...But if any one receive not, as sacred and canonical, the said books entire with all their parts, as they have been used to be read in the Catholic Church, and as they are contained in the old Latin vulgate edition; and knowingly and deliberately contemn the traditions aforesaid; let him be anathema.
