The official name of the "Mormon Church" is The Church of Jesus Christ of Latter-day Saints. Mormons believe in Jesus Christ and celebrate Christmas in recognition of Jesus' birth. Like many other Christians, we also feel that Christmas is a good time to remember Jesus' life and atonement for us.
Mormon families that I've experienced Christmas with tend to read the nativity story in Luke 2 on Christmas Eve or Christmas day. I also like to read the Book of Mormon chapter 3 Nephi 1, which describes what happened in the Americas at the time of Christ's birth.
There's no special religious service that Mormons attend on Christmas Eve or Christmas day, unless one of those days falls on a Sunday. In that case Sacrament Meeting is held as usual, perhaps with a few special musical numbers added to the service. If Christmas day is a Sunday, there are usually no other church meetings held besides Sacrament Meeting so that members can spend time with their families.
Mormons also participate in secular Christmas traditions such as exchanging presents, talking about Santa Claus, lighting fireworks (if you live in Latin America), etc. However, church leaders frequently urge members to remember the spiritual side of Christmas ahead of the secular traditions.
The first Sunday of December the First Presidency of the Church holds a special Christmas devotional in Salt Lake. The music is superb, with the Mormon Tabernacle Choir and the Orchestra at Temple Square performing several numbers. You can find out more about how to watch the devotional here.