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):
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:
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.
The exact day for the birth of Jesus is not known. You can find a lot of info on Wikipedia, with references. The most interesting for this question might be: