The Nativity in the Gospels

The birth of Christ is not mentioned in the Gospels of Mark and John. There is a brief description in the Gospel of Matthew (in the Latin Vulgate, translated by St. Jerome). Here is how it reads in the literal Douay-Rheims translation, after several verses listing the genealogy of Jesus (tracing his lineage back to King David):

16 And Jacob begot Joseph the husband of Mary, of whom was born Jesus, who is called Christ. 17 So all the generations, from Abraham to David, are fourteen generations. And from David to the transmigration of Babylon, are fourteen generations: and from the transmigration of Babylon to Christ are fourteen generations. 18 Now the generation of Christ was in this wise. When as his mother Mary was espoused to Joseph, before they came together, she was found with child, of the Holy Ghost. 19 Whereupon Joseph her husband, being a just man, and not willing publicly to expose her, was minded to put her away privately. 20 But while he thought on these things, behold the angel of the Lord appeared to him in his sleep, saying: Joseph, son of David, fear not to take unto thee Mary thy wife, for that which is conceived in her, is of the Holy Ghost.

21 And she shall bring forth a son: and thou shalt call his name JESUS. For he shall save his people from their sins. 22 Now all this was done that it might be fulfilled which the Lord spoke by the prophet, saying: 23 Behold a virgin shall be with child, and bring forth a son, and they shall call his name Emmanuel, which being interpreted is, God with us. 24 And Joseph rising up from sleep, did as the angel of the Lord had commanded him, and took unto him his wife. 25 And he knew her not till she brought forth her firstborn son: and he called his name JESUS.

However, most of the infancy narrative so familiar to us as the Christmas Story is found in the second chapter of the Gospel of Luke (in the Latin Vulgate). Here is how it reads in the Douay-Rheims translation:

1 And it came to pass, that in those days there went out a decree from Caesar Augustus, that the whole world should be enrolled. 2 This enrolling was first made by Cyrinus, the governor of Syria. 3 And all went to be enrolled, every one into his own city. 4 And Joseph also went up from Galilee, out of the city of Nazareth into Judea, to the city of David, which is called Bethlehem: because he was of the house and family of David, 5 To be enrolled with Mary his espoused wife, who was with child.

6 And it came to pass, that when they were there, her days were accomplished, that she should be delivered. 7 And she brought forth her firstborn son, and wrapped him up in swaddling clothes, and laid him in a manger; because there was no room for them in the inn. 8 And there were in the same country shepherds watching, and keeping the night watches over their flock. 9 And behold an angel of the Lord stood by them, and the brightness of God shone round about them; and they feared with a great fear. 10 And the angel said to them: Fear not; for, behold, I bring you good tidings of great joy, that shall be to all the people:

11 For, this day, is born to you a Saviour, who is Christ the Lord, in the city of David. 12 And this shall be a sign unto you. You shall find the infant wrapped in swaddling clothes, and laid in a manger. 13 And suddenly there was with the angel a multitude of the heavenly army, praising God, and saying: 14 Glory to God in the highest; and on earth peace to men of good will. 15 And it came to pass, after the angels departed from them into heaven, the shepherds said one to another: Let us go over to Bethlehem, and let us see this word that is come to pass, which the Lord hath shewed to us.

16 And they came with haste; and they found Mary and Joseph, and the infant lying in the manger. 17 And seeing, they understood of the word that had been spoken to them concerning this child. 18 And all that heard, wondered; and at those things that were told them by the shepherds. 19 But Mary kept all these words, pondering them in her heart. 20 And the shepherds returned, glorifying and praising God, for all the things they had heard and seen, as it was told unto them.

These are the only accounts of Christ's birth in the four Gospels. In harmony with this, the feast of Christmas was not particularly important in the early church, overshadowed by a focus on Easter. There is a good account of the development of Christmas in this article from The Catholic Encyclopedia:

Christmas was not among the earliest festivals of the Church. Irenaeus and Tertullian omit it from their lists of feasts; Origen, glancing perhaps at the discreditable imperial Natalitia, asserts (in Lev. Hom. viii in Migne, P.G., XII, 495) that in the Scriptures sinners alone, not saints, celebrate their birthday; Arnobius (VII, 32 in P.L., V, 1264) can still ridicule the "birthdays" of the gods. [...]

The crib (creche) or nativity scene. St. Francis of Assisi in 1223 originated the crib of today by laicizing a hitherto ecclesiastical custom, henceforward extra-liturgical and popular. The presence of ox and ass is due to a misinterpretation of Isaias i:3 and Habacuc 3:2 ("Itala" version), though they appear in the unique fourth-century "Nativity" discovered in the St. Sebastian catacombs in 1877. The ass on which Balaam rode in the Reims mystery won for the feast the title Festum Asinorum (Ducange, op. cit., s.v. Festum).

Hymns and carols. The degeneration of these plays in part occasioned the diffusion of noels, pastorali, and carols, to which was accorded, at times, a quasi-liturgical position. Prudentius, in the fourth century, is the first (and in that century alone) to hymn the Nativity, for the "Vox clara" (hymn for Lauds in Advent) and "Christe Redemptor" (Vespers and Matins of Christmas) cannot be assigned to Ambrose. "A solis ortu" is certainly, however, by Sedulius (fifth century). The earliest German Weihnachtslieder date from the eleventh and twelfth centuries, the earliest noels from the eleventh, the earliest carols from the thirteenth. The famous "Stabat Mater Speciosa" is attributed to Jacopone da Todi (1230-1306); "Adeste Fideles" is, at the earliest, of the seventeenth century. These essentially popular airs, and even words, must, however, have existed long before they were put down in writing.

The verse mentioned in that article ("The ox knoweth his owner, and the ass his master's crib: but Israel hath not known me, and my people hath not understood.") from Isaiah 1:3 made the ox and donkey part of Christmas imagery before St. Francis, too. The other verse mentioned (Habakkuk 3:2) was sometimes mistranslated, in the Greek Septuagint, as "In the midst of two animals thou shalt be known." St. Jerome, working from the original Hebrew, translated that verse correctly as "In the midst of years."