Released in 1898, Santa Claus is believed to be the first movie with a Christmas theme. If it is not, it appears to be the earliest extant Christmas movie. It is also the first film to depict Santa Claus. Running under two minutes, it simply depicts two children going to bed, Santa coming down their chimney, leaving them presents in their stockings, then leaving, and the children finding their surprise.
Filmed in England, this was one of the most sophisticated films made at that time. One thing to realize is that electric lights weren't strong enough to light the set: the bedroom is a set that was actually outside. To simulate darkness, a curtain is drawn over the bedroom background (you can see a jump cut). The image of Santa going down the chimney appears to be a clever double exposure work.
Because that runs at two minutes, here's a couple longer silents.
First is The Night Before Christmas from 1905. Produced by the Edison Company, the classic poem is re-enacted with a little expansion.
And from 1914, here's The Adventure of the Wrong Santa Claus. Octavius, Amateur Detective (who had starred in a series of silent comedies, of which this was the last) visits his friend in costume as Santa Claus, only to be knocked out by a thief also dressed as Santa who robs the family. Octavius must catch the thief!