In the conventional version of the Christmas story, the wise men or magi:
- Melchior and
started the gift-giving custom of Christmas by bringing gold, frankincense, and myrrh to the Christ child on Epiphany, the day on which the infant was presented. The 3 magi have been described not only as wise men, but also as kings or Persian priests and astrologers.
Epiphany is the end of the Christmas season, 12 days after Christmas, which is, literally, the mass for Christ
Other Names for the MagiThe magi were given other names, as well, including Apellus, Amerus and Damasius, which were used in Peter Comestor's medieval Historia Scholastica (source: "A Lexicon of Christian Iconography," by Bruce M. Metzger; Church History, Vol. 45, No. 1 (Mar., 1976), pp. 5-15.).
Christ + Mass = Christmas
Christmas is often celebrated the evening before Christmas day, and Epiphany is often celebrated as the Twelfth Night. Gift-giving in some cultures extends throughout the 12 days of Christmas and in some places is limited to January 5 or 6. Similarly, for those who celebrate only Christmas, gifts are exchanged on either December 24, Christmas Eve, or December 25, Christmas Day. Many Orthodox Christians celebrate Christmas on January 7 because of the difference between the Gregorian and Julian calendars.
Most Familiar Reference to the Magi
Matthew mentions but neither numbers nor names the wise men:
 Now when Jesus was born in Bethlehem of Judaea in the days of Herod the king, behold, there came wise men from the east to Jerusalem,
 Saying, Where is he that is born King of the Jews? for we have seen his star in the east, and are come to worship him.
 When Herod the king had heard these things, he was troubled, and all Jerusalem with him.
 And when he had gathered all the chief priests and scribes of the people together, he demanded of them where Christ should be born.
 And they said unto him, In Bethlehem of Judaea: for thus it is written by the prophet,  And thou Bethlehem, in the land of Juda, art not the least among the princes of Juda: for out of thee shall come a Governor, that shall rule my people Israel.
 Then Herod, when he had privily called the wise men, inquired of them diligently what time the star appeared.
 And he sent them to Bethlehem, and said, Go and search diligently for the young child; and when ye have found him, bring me word again, that I may come and worship him also.
 When they had heard the king, they departed; and, lo, the star, which they saw in the east, went before them, till it came and stood over where the young child was.
 When they saw the star, they rejoiced with exceeding great joy.
 And when they were come into the house, they saw the young child with Mary his mother, and fell down, and worshipped him: and when they had opened their treasures, they presented unto him gifts; gold, and frankincense, and myrrh.
 And being warned of God in a dream that they should not return to Herod, they departed into their own country another way.
 And when they were departed, behold, the angel of the Lord appeareth to Joseph in a dream, saying, Arise, and take the young child and his mother, and flee into Egypt, and be thou there until I bring thee word: for Herod will seek the young child to destroy him.
 When he arose, he took the young child and his mother by night, and departed into Egypt:
 And was there until the death of Herod: that it might be fulfilled which was spoken of the Lord by the prophet, saying, Out of Egypt have I called my son.
 Then Herod, when he saw that he was mocked of the wise men, was exceeding wroth, and sent forth, and slew all the children that were in Bethlehem, and in all the coasts thereof, from two years old and under, according to the time which he had diligently inquired of the wise men.
King James Version of Matthew 2
[See: A Medieval Christmas for more information on Christmas customs.]