We can thank Irenaeus, not only for identifying the tetramorph gospel, but also for providing the names of the authors who are traditionally assigned to them. He summarized the tradition he received:
Matthew also issued a written Gospel among the Hebrews in their own dialect, while Peter and Paul were preaching at Rome, and laying the foundations of the Church. After their departure, Mark, the disciple and interpreter of Peter, did also hand down to us in writing what had been preached by Peter. Luke also, the companion of Paul, recorded in a book the Gospel preached by him. Afterwards, John, the disciple of the Lord, who also had leaned upon His breast, did himself publish a Gospel during his residence at Ephesus in Asia.—Against Heresies, Book 3, Chapter 1.
Irenaeus was likely dependant on Papias (who Eusebius quotes):
But concerning Matthew [Papias] writes as follows: "So then Matthew wrote the oracles in the Hebrew language, and every one interpreted them as he was able."—The Church History of Eusebius, Book 3, Chapter 39.
From a historical perspective, this is a bit awkward as the evidence is that Matthew was originally written in Greek and not Hebrew. But for the purposes of this question, that's the answer. (See also: Gospel of Matthew on the Early Christian Writings site.)
Papias gives also in his own work other accounts of the words of the Lord on the authority of Aristion who was mentioned above, and traditions as handed down by the presbyter John; to which we refer those who are fond of learning. But now we must add to the words of his which we have already quoted the tradition which he gives in regard to Mark, the author of the Gospel.
This also the presbyter said: Mark, having become the interpreter of Peter, wrote down accurately, though not in order, whatsoever he remembered of the things said or done by Christ. For he neither heard the Lord nor followed him, but afterward, as I said, he followed Peter, who adapted his teaching to the needs of his hearers, but with no intention of giving a connected account of the Lord’s discourses, so that Mark committed no error while he thus wrote some things as he remembered them. For he was careful of one thing, not to omit any of the things which he had heard, and not to state any of them falsely.
These things are related by Papias concerning Mark.—The Church History of Eusebius, Book 3, Chapter 39.
So Eusebius quotes Papias (of whose works we only have fragments quoted in other works) quoting1 John the Presbyter who identifies Mark as recording Peter's gospel message. Although Irenaeus also identifies Mark as the author, it's likely he got his information from Papias. Justin Martyr seems to2 also link the Markian text with Peter, which is an independent confirmation. Usually this Mark is identified with John Mark mentioned a few times in Acts and one of Paul's letters. There really aren't any other candidates if you credit Papias' account. (See also: Gospel of Mark on the Early Christian Writings site.)
Appropriately, the author of the earliest church history (Acts) and the most careful study of Jesus' life among the Gospels (Luke) is fairly conclusively identified with Luke the physician. Clement of Alexandria positively identifies Luke as the author of Acts:
It remains that we understand, then, the Unknown, by divine grace, and by the word alone that proceeds from Him; as Luke in the Acts of the Apostles relates that Paul said, “Men of Athens, I perceive that in all things ye are too superstitious. For in walking about, and beholding the objects of your worship, I found an altar on which was inscribed, To the Unknown God. Whom therefore ye ignorantly worship, Him declare I unto you.”—The Stromata, Book 5, Chapter 12.
Clement also quotes a bit of Luke's gospel and names him as the author in another book. Tertullian and the earliest manuscript bear witness to Luke's authorship. Since the recipient of the work is included in the preface, it's even possible the author signed the copy he originally submitted, though we don't actually have that copy to know for sure. (See also: Gospel of Luke on the Early Christian Writings site.)
Internal evidence suggests that author of John was the Beloved Disciple. Traditionally, that figure has been associated with John, one of Jesus' inner circle. Again, Irenaeus likely received his information from Papias. Eusebius jumps quickly over the question of who wrote the Gospel of John (probably because the identification was common knowledge by the 3rd-century) in order to address the authorship of Revelation:
It is worth while observing here that the name John is twice enumerated by him. The first one he mentions in connection with Peter and James and Matthew and the rest of the apostles, clearly meaning the evangelist; but the other John he mentions after an interval, and places him among others outside of the number of the apostles, putting Aristion before him, and he distinctly calls him a presbyter.
This shows that the statement of those is true, who say that there were two persons in Asia that bore the same name, and that there were two tombs in Ephesus, each of which, even to the present day, is called John’s. It is important to notice this. For it is probable that it was the second, if one is not willing to admit that it was the first that saw the Revelation, which is ascribed by name to John.—The Church History of Eusebius, Book 3, Chapter 39.
By calling John the apostle, "the evangelist", Eusebius identifies him as the author of the gospel (euangelion in Greek). He proposes another John as the author of Revelation.2 There are a number of other candidates, but this is how John got associated with the gospel that bears his name. (See also: Gospel of John on the Early Christian Writings site.)
Or perhaps paraphrasing. Ancient texts didn't use much in the way of punctuation, so it's hard to delineate quotations in the best of circumstances. With this many layers of quotation, it's especially rough.
The key is to see that Justin Martyr mentions Boanerges, which indicates the Gospel of Mark. He calls the gospels the "memoirs of the apostles" and Mark seems to be Peter's memoir:
And when it is said that He changed the name of one of the apostles to Peter; and when it is written in the memoirs of [him] that this so happened..."—Dialogue of Justin, Philosopher and Martyr, with Trypho, a Jew, Chapter 106.
However, the memoir could be of Jesus ("Him"), so this is only supplementary evidence.
It's possible that this is a misreading of Papias by Eusebius. Irenaeus seems to think there was just one John in the Asian church:
And these things are borne witness to in writing by Papias, the hearer of John, and a companion of Polycarp, in his fourth book; for there were five books compiled (συντεταγμένα) by him.—Against Heresies, Book 5, Chapter 33.
Perhaps at issue is the authorship of Revelation, which was the source of the early chiliasm controversy. If there are two authors named John (one for the Gospel and one for the Apocalypse), than one may be taken as historical and the other as fanciful.