A quarter is one-fourth of something. If you cut a pizza into four pieces and share it with three friends, each of you will eat a quarter of the pizza.
A quarter is the same thing as twenty-five percent of something. Many sporting events are divided this way, and you'll often hear sports commentators saying things like, "Here is the score at the end of the first quarter." The Latin word quartus, "fourth," is at the root of quarter, and the earliest — and goriest — use specifically referred to a form of execution that involved dismembering a prisoner's body into four pieces, or quarters.