Some historical background regarding the Canon of Scripture
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).
Regarding the O.P.'s original questions
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.