What is that thing called that is a sofa like thing but fold down into a bed?
I think what you're looking for is a FUTON. With a futon sofa, the back of the sofa that you lean against while sitting folds down flat to make a bed. A sofa bed, on the other hand, looks like a regular sofa, but you remove the bottom seat cushions to pull the bed out, which folds out from under where you generally sit. A day bed, as someone suggested, is simply a twin bed that pushes up against the wall on the long side, with the headboard running the length of the bed (whereas, on a normal twin bed, the headboard is narrow, and sits at the head of the bed).