It's January 6, Feast of the Wise Men, also known as Epiphany (from the Greek, meaning a vision or revelation). All self-respecting manger scenes today have both shepherds and kings (wise men) bearing gifts. The Bible version is a little more complicated.
The book of Matthew is the only one of the gospels to mention the wise men. They come not to a manger but to Mary and Joseph's house in Bethlehem, where they have lived since their marriage. They are not called kings, just wise men (magi), and their number is unspecified. The assumption that there were three of them is doubtless due to their bringing three gifts, gold, frankincense, and myrrh.
These three gifts were intended to illustrate Christ's three roles, as king (the gold), as priest (the incense), and as sacrifice (myrrh was used in embalming). He both orders and performs the sacrifice and is the sacrifice Himself.
In Matthew, King Herod is not happy to hear about a baby born to be King of the Jews and orders all baby boys under the age of two to be killed. The Feast of the Innocents, commemorating the slaughter of the baby boys, is usually celebrated on December 28, suggesting that in medieval theology (when all these dates were chosen) it took Herod close to a year to figure out that the wise men who had promised to come tell him where the baby lay (they'd arrived on January 6) weren't coming to tell him about it after all. In the Bible, they are warned in a dream not to say anything to Herod.
Matthew has Mary and Joseph flee to Egypt to escape Herod, then settle in Nazareth once they return, rather than going back to Bethlehem. This is different from Luke (the only gospel to mention shepherds), which has Mary and Joseph living in Nazareth the whole time, but going to Bethlehem to be enrolled/taxed and finding no place to stay but a stable. Here there is no mention of Herod and no flight into Egypt. Both of these gospels have Jesus born in Bethlehem and grow up in Nazareth, which fulfills Old Testament prophecies, but they give different explanations for how it happened.
The Feast of the Wise Men is sometimes called Twelfth Night, which doesn't quite work. If you count Christmas Day as the first day of Christmas, then the twelfth would be January 5. It is however twelfth night, which would mean it could begin at sundown on the 5th (using the Jewish version that days begin at the previous sundown) and include the 6th. The "eve" is important to a number of religious holidays, including Hallowe'en and Christmas Eve (but if you start counting Christmas from sundown on Christmas Eve, you still don't get to January 6). Maybe one should count the "twelve days of Christmas" as starting the day after Christmas--more time for food and fun (note: there are no partridges or pear trees in the New Testament).
In Italy, children receive gifts on Epiphany. They are not brought by Santa but by Befana, a wicked fairy (try to guess what word her name is corrupted from--hint, put the letter E at the beginning of Befana, before the B, and say it out loud). The story is that the Wise Men asked her where to find baby Jesus and she refused to answer. After they left, however, she got to thinking that maybe she should have gone with them, and she's spent the last 2000 years wandering around giving children gifts, in the hope that one of them might be the Christ child.
December 6, Feast of Saint Nicholas, and January 6, Feast of the Wise Men. A whole month of opportunities for presents, especially if you include Hanukkah, New Year's (when Romans and medieval people exchanged gifts), and, why not? Kwanzaa. Parents don't seem to understand their children's logic on this point.
For more on the history of Christmas, check out my book "Contested Christmas," available from Amazon and other online bookstores.