The dog's name in "How the Grinch Stole Christmas" is Max. He is the Grinch's loyal, long-suffering pet and companion.
Max first appeared in Dr. Seuss's 1957 picture book, where the Grinch recruits him to help carry out the Christmas theft. The Grinch disguises himself as Santa Claus, while Max is dressed up as a reindeer and helps pull the sleigh through Whoville. Despite being treated poorly, Max remains devoted to his owner.
Max also appears in the famous 1966 animated television special and the later film adaptations, including the 2000 live-action movie starring Jim Carrey and the 2018 animated film. His role stays broadly the same: he assists the Grinch but also provides a sympathetic contrast to the Grinch's bitterness.
A common mix-up is confusing Max with other animal characters associated with Christmas stories, such as Santa's reindeer or the unnamed animals in other Dr. Seuss works. Max is specifically the Grinch's dog; when he wears antlers, he is only pretending to be a reindeer.