The traditional Japanese futon was created to save space, as many modern futons are made to do today. This was because many Japanese families only had one room to live in. This was nothing more than a thin mattress laid only on the tatami mat floor of their home. They would store it during the day.
This is the most common type of futon in the West. In this style, the mattress is folded in half on top of the frame. The frame then slides in to a flat position for sleeping. Bifold futons can come with different sizes of mattress, affecting the size of the couch when it is folded. As the mattress has to go through constant folding and unfolding, it is usually made from synthetic material and cotton.
This type of futon is designed to fit in a very small space, and can only fit two sizes of mattress: full and twin. As well as acting as a bed and loveseat, loveseat futons also have a third "lounger" position, in which a small deck pulls out to elevate the legs. This puts the futon somewhere in between the two positions of bed and sofa. The advantage of this type of futon is that the mattress can be thicker than that on a trifold futon without taking up more room.
These are also built to save space. Only thin mattresses can go on a trifold futon so the mattress can properly fold over the back of the frame in couch position. These mattresses are often less comfortable due to their thinness, and are not recommended for regular use.