A rise is the indentation created when a fish (usually a trout) breaks or almost breaks the surface when it is feeding.
Rises occur during a hatch when the fish are feeding on insects that are in the surface film or just below.
Technically a fish leaping out of the water to feed on flying insects such as mayfly spinners or damselfies is a form of rise too.
Generally, though, rises fall into two categories: the classic dry fly rise and the emerger rise.
The classic dry fly rise sees the fishes snout actually break the surface of the water creating a dimple and then concentric rings emerging out of where the rise took place.
The emerger rise occurs when the fish is able to eat the mayfly (or caddis or whatever hatching insect) before it breaks through the surface film. This rise appears as a bulge as the snout of the rish doesn’t actually break the surface.
There is more info here on trout rise forms and in the video below.