Some people do celebrate on December 25th, but not Christmas but Soorb Stepanos Day (St Steven's day). In most other countries St Stephen's day is on December 26th or Boxing Day!
Santa Claus Gaghant Baba / Kaghand Papa traditionally comes on New Year's Eve (December 31st) because Christmas Day itself is thought of as more of a religious holiday in Armenia.
In Armenian Happy/Merry Christmas is 'Shnorhavor Soorb Tsnund' (which means 'Congratulations for the Holy Birth'). Happy/Merry Christmas in lots more languages.
At the beginning of December, a big Christmas Tree (Tonatsar) is put up in Republic Square in Yerevan, the capital of Armenia.
Favourite and traditional Holiday foods in Armenia include Anooshaboor (Armenian Christmas Pudding), Khozee bood (glazed ham) and dried fruits. Every house is ready with lots of sweets because anyone might knock on the door and come in for a party!