The box spring used in traditional beds resembles a mattress but has a supportive structure and several springs inside, hence the name "box springs." The box spring sits on the inside edge of the metal bed frame and creates the surface the mattress sits on. The springs are designed to give slightly, making the mattress less firm than if on a platform.
A platform bed consists of either slats or a sturdy, solid piece of wood or plywood laid on a frame or as the top of the frame. The mattress sits directly on top of this sturdy layer, eliminating the need for a box spring. Certain mattresses need the solid surface as the materials they are made of are not meant to give. Waterbed mattresses also lay on top of a platform frame; there is nothing between the frame and the water-filled mattress, making a sturdy, solid, non-giving surface.
No box spring required can save you hundreds of dollars -- or more -- when purchasing a mattress. When shopping for mattresses, oftentimes the price says "set," which means the set of box spring and mattress. The platform bed also sits lower to the ground than a typical frame; this can be a pro or con depending on which you like better. The platform bed also lends more to contemporary and modern designs in the bedroom.
When making the choice between the two styles of frames, try them both before choosing to give yourself a more educated purchase. If you're buying a specialty mattress or one that comes with a warranty, make sure that the type of frame you choose will not void the warranty (you may have to call the manufacturer specifically; don't always rely on the salesperson).