Why is Christmas on December 25th? Why was December 25th chosen as the day to celebrate Jesus' birth?
This is a good recent blog post covering Pope Benedict's thoughts on the matter.

The key is the date set for the annunciation (March 25th):


Traditionally held to be the first day of creation
Traditionally held to be the date when Abraham was to sacrifice Isaac after a 3 day journey to Mount Moriah
Extrapolated to be the date of Jesus' annunciation (when the Angel Gabriel came to Mary)
Futher extrapolated to be the date of Jesus' crucifixion (on the Lunar calendar at least) to parallel Abraham's attempted sacrifice and align with the passover. 


Therefore, as vsz says wikipedia says, but should say louder, add 9 months, et voila you've got December 25th.

As for the age of the feast as celebrated by Christ's Church, said blog post also notes St. Pope Leo the Great mentioning the feast of the Nativity of Jesus in the winter:


  But this Nativity which is to be adored in heaven and on earth is suggested to us by no day more than this when, with the early light still shedding its rays on nature, there is borne in upon our senses the brightness of this wondrous mystery. (St Leo Magnus, Sermo 26)




I'm not going to say that only stupid people think that Christmas is an attempt to quash pagan solstice shenanigans, but I will say that people ought to think long and hard before believing the Discovery channel over thousands of years of tradition.