How many types of pistons are there they are two types of piston?

There are many, many types of pistons. The pistons of a 4-cycle gasoline internal combustion engine are different from those in a diesel powered or two-cycle two cycle engines. There are other alternative fuels that require specialized pistons too. Then within a group, such as 4-cycle gasoline automobile engines there can be domed, or pistons with valleys or cut-outs for the valves. There short sleeves, long sleeves, thick or thin wall, or cuts for different types of rings. There are literally thousands of different types or variations of pistons.