What is the process by which the Gospels entered the canon?

If you have read the gospels of Matthew, Mark, Luke, and John, you will know that in many cases, they repeat the same stories as each other. There are far more books—called "apocrypha"—that were circulated among the early Christians, also containing broadly similar material, which did not make it into the final, accepted version of the New Testament (the "canon"). We know, however, that the key four gospels, or "Tetramorph," were already generally accepted by 180 AD, when they are referred to by Ireneaus in his writings; the Pauline epistles were also in wide circulation by this point.
By the middle of the third century AD, it is thought that the twenty-seven books of the modern New Testament were widely agreed upon. The process by which this agreement was written down involved a number of church councils, with the Council of Rome famous as key among these. Athanasius, Bishop of Alexandria, was the first to use the word "canon" in relation to the twenty-seven books, which he set out in a letter written at Easter in the year 367. By the time of St. Augustine, the matter of the New Testament canon was regarded as settled, although debate continued over the Book of Revelation and other later sections of the Testament. The gospels themselves, however, were some of the first sections of the Testament to be agreed upon as canon. These had been circulated and debated in the very early church in Rome, and effectively the books considered closest in time to Jesus were accorded privilege; any gospel that appeared to tell contradictory stories was quickly weeded out, although some of these seeded heretical sects which would survive for centuries.