Pool coping can be divided into four main types. Rolled coping, cantilevered, bull-nosed, and rough-cut.
Rolled edge coping is made from brick, concrete, or cast stone. It is smooth and upturned at the edge which can give a swimmer something to grab onto.
Cantilevered coping is straight on top, runs out past the edge of the pool, then drops down a bit, forming a 90 degree angle. It offers a modern look, and the L shape makes the deck material seem thicker than it actually is.
Bull-nosed coping is made from stone, cast-stone, or from cast-concrete edges. It is flat with a rounded end.
Rough-cut coping is good for giving the pool a casual look. It is made from rough stone and has a textured edge.
Instead of stone or concrete, you can have pool coping done in tile as well. Tile is a good choice, as well, to extend down below the waterline a bit. This makes cleaning easier as dirt and scum that lurks at the waterline is simple to brush off smooth tile (but with concrete it can linger in the nooks).