Any tree of genus Malus which does not grow apples.
Now does that make any sense? Well, it turns out that the only distinction between apples and crabapples is the size of the fruit - if they're greater than 2 inches in diameter, they're apples and the tree they're growing on is an apple tree. If they're that size or smaller, they're crabapples and the tree is a crabapple tree. There is no deeper meaning behind the distinction.
Being that small, crabapples aren't often eaten, although they can be made into jelly (and perhaps even wine if you're good at that sort of thing). The fruit are very sour and often referred to as "crab apples," unlike the tree which is usually written as a single word.
Crabapple trees are most often not grown for their fruit. Malus is part of family Rosaceae - the roses - and the trees are typically grown for ornamental purposes, producing colorful flowers. Most of those that have been bred specifically for their flowers have small fruit, often less than an inch in diameter and referred to as "berries" instead of apples. Birds like them.
When I was a kid, my grandparents lived next to an old lady who had a crabapple tree in her back yard. These were rowhouses, so there wasn't much room, and the tree's boughs hung over the fence. The little apples would ripen and fall to the ground. They made excellent slingshot ammunition, hard enough to fly straight and give someone a good knock on the noggin, but yielding enough to avoid drawing blood. Passing motorists didn't seem to appreciate it when a shot sailed high, though.