There are a variety of typical mattress sizes that are available depending on your personal size and whether you share a bed with another person. Generally the smallest size of mattress is a single at 39" x 75". Moving up in size are double mattresses at 54" x 75". The classic mattress size for a shared bed is a "Queen" at 60" x 80". The larger mattresses are "King" sized at 76" x 80".
The most common form of mattress is the spring coil which was invented sometime around the later 19th century. The springs provide a flexible support for the body as it lies on the otherwise shapeless cloth of the mattress. Modern mattresses usually use "inner-spring" coils, which are made of steel and combined together inside the mattress in a variety of degrees of tightness, depending on how hard the desired mattress is.
The classic inner-spring mattress will likely be the standard bedding for the foreseeable future, but manufacturers have also created a variety of other types of mattress that take advantage of newer materials and design. Air mattresses, which simply use compressed air instead of springs to provide support, have proven to be useful as portable bedding. Foam mattresses, of the sort developed by NASA, provide support for a body without being quite as hard as springs.