Top mount or Drop in sinks sit on top of the countertop surface. They usually have a small rim that is visible around the sink. Faucet holes and sink accessory holes are pre-drilled into this sink mounting piece.
Undermount sinks are attached to the countertop underneath. The sink sits below the surface and the cut out for the sink is visible. This makes the cut out for the sink a finished edge.
Vessel sinks have the appearance of sitting on or above the countertop. The cut out for vessel is a simple hole for the drain or a small cut out for part of the sink.