As an addendum to weasello's excellent writeup above, and for contrast, I offer the following list of aircraft classes as they are (or soon will be) recognized by the U.S. FAA:

Note that aircraft must be certified to be flown in the U.S., with the exception of experimental aircraft. Recognizing the continuing debt owed to the garage tinkerer by general aviation (and, by extension, everything up to and including the Space Shuttle - thanks Orville and Wilbur!) the FAA allows homebuilt and even home designed airplanes restricted use of the nation's airspace. The tradition of experimental aircraft designs flown by their inventors is going strong - see Bert Rutan for an example.