A cusp in geometry is the point where two curves meet. It's a kind of transition. If you're on the cusp of manhood, you’re not quite grown up, but you’re definitely not a little boy anymore either.
The word comes from Latin cuspis, "point." You'll see lots of cusps in Gothic churches — a point where two arches meet, or where decorative curves meet to form a point. The little points that stick out on the surface of your teeth that are flat on top — the ones you use for grinding — are also called cusps. Those teeth right behind your canine teeth but before your molars have two cusps, and they're called bicuspids.