A daybed is a kind of bed similar to a futon, in that it functions as a sofa/bed combination. They have three pieces: two sides and a back panel, and are built in a wood/metal hybrid, if not just one or the other. They're built to hold a twin size mattress, and can be used in apartments, guest rooms, dens, offices, sun rooms, you name it. Anywhere you could possibly see an excuse to have a sofa/bed and look classy, daybeds are your pick. Where futons are typically better couches or sofas, daybeds are better beds.
A link spring is the common support system for the mattress, which is a metal frame roughly the size of the mattress, only flat, with cross supports. Wire, polyester or nylon meshes are held in place by springs which also allows room underneath the daybed for storage. Some of these will include an additional bed underneath.