If a small flake of your purple nail polish breaks off, you might describe it as a chip, or a tiny piece. You might also say, "Shoot, I always seem to chip my nails."
You can use the word chip as a noun or a verb, to describe the breaking off of a small piece or the small piece itself. You might accidentally chip your aunt's favorite tea cup, or realize that slamming your door made a chip of paint fall off your bedroom wall. The word comes from the Old English forcippian, "to pare away by cutting," and the related cipp, also pronounced chip, which means "small piece of wood."