In Christianity, Christmas is the celebration of the birth of Jesus. Let's leave aside for now any discussion of the historicity of Jesus and simply assume the Biblical account of his birth is correct. Is there scriptural evidence that Jesus was born on December 25? The short answer is no. Shepherds weren't out in the fields in December. They were out March through October. If Jesus died at 33 1/2 years old, on Passover, his birthdate would have been in mid-September. A Google search will clearly reveal that a number of people (some better qualified to do so than others) argue rather strongly for one date or another, but no one seriously argues for December 25 (and he was likely born sometime around 2-5 BC as well).
Why do we celebrate on December 25? Because once the early Christian church became powerful enough, it sought to stomp out all the old pagan beliefs and mid-December was a time of feasting and celebration in ancient Europe. It's when the winter solstice occurs, the turning of the year, and the church couldn't have people engaged in Sun worship so, in 350, Pope Julius I declared that Christ’s birth would be celebrated on December 25. Many of the old pagan traditions were hard to stomp out, however, and carried over into our modern Christmas celebrations.
Holly, ivy, and mistletoe are all plants associated with pagan mid-winter celebrations. Mistletoe has an interesting cultural history. European mistletow grows as a semi-parastitic plant on oak trees (sacred to the druids) and remains as a green crown on the seasonally "dead" trees in winter. It was seen as a powerful fertility symbol. The waxy white berries were identified with drops of semen - think about that next time you kiss under the mistletoe. I've read that mistletoe was banned from early Christian churches due to its pagan associations.
Yule logs are another of the pagan holdovers (a Norse tradition). So are wreaths and cuttings of evergreens (Germanic). Pagans did not erect Christmas trees (probably because they wouldn't cut down a living tree and bring it into a house) but there is a long pagan history of decorating living trees. As an aside, some fundamentalist Christians refuse to have a Christmas tree in their home because of Jeremiah 10:1-5.
While the Santa Claus legend is usually attributed by Christians to St. Nicholas of Myra in the 4th century, Santa has a lot of parallels to the old Germanic/Norse legends of Odin. Around Yule, Odin would lead a great hunt riding an 8-legged horse named Sleipnir that could leap great distances (kind of like magical reindeer). Odin also was referred to as having a long beard. Children would place boots filled with carrots or straw near the fireplace for Sleipnir to eat and Odin would leave gifts in return (like hanging stockings by the fire).
I'm half Finn and in Finland Santa is called Joulupukki which literally means "Yule Goat". This likely comes from an old pagan tradition of evil spirits that wore goat skins and horns and came in the night to demand gifts. This later morphed into a more jolly figure. Joulupukki is where we get the reindeer, the red and white suit, the helper elves (Joulutonttu), and the cold North Pole origin (Lapland in Finland originally).
Bottom line is that what we call Christmas is a mish-mash of European traditions, both Christian and pagan. So, despite what you hear in church, Christ is not the only reason for the season (I think he'd be horrified by the greed and commercialism of the holiday anyway - see Matthew 21:12). The tilt of the Earth's axis in the European mid-latitude climate is the real reason we have a seasonal celebration - it's ultimately astronomical!