There are basically two types of seats: Seats with rails and seats without rails.
Seats with rails (two somewhat crooked metal bars running front-to-back under the seat) will fit the clamp in Mathew's photo or any fancier seat clamp. (For fancier ones generally the clamp is permanently attached to the top of the seat post (the part that slides up and down)).
Seats without rails generally have a built-in clamp, or in some cases may be more or less permanently attached to the seat post. These are "cost-reduced" seats, designed for cheap bikes and exercise equipment. They are not generally interchangeable without replacing the entire assembly.
You can replace the "without rails" seat with one that has rails if you get the necessary clamp or clamp-with-seatpost. The clamp will accept any brand/style of seat with rails, but there are several different diameters of seat posts, and you need to match the post diameter.
If your current seat doesn't have rails, your best bet is to take the entire seat assembly -- seat, clamp, and post -- to a bike shop, so that the seat post diameter can be matched.