Camera formats are the same as film formats. So, a camera is referred to by the film it uses. The same categories apply as for film.
These are Minox-type cameras, with 8 x 11mm negative, and those that use the 110 cassette. The Minox format has gained cult status (getting a print from that tiny negative takes some patience), and the film for these cameras is still being produced in limited quantities. 110 was particularly popular early in its introduction in the 1970's for promotional and novelty cameras, as well as consumer use. Lomography still markets both cameras and film.
Small format uses 35mm (135) roll film. This category is dominated by the 35mm Single Lens Reflex (SLR) camera, but also includes rangefinders and point-and-shoots. These cameras are all compact, and the format is large enough to make good-sized prints. It is very adaptable format, combining relatively small-sized equipment (both the cameras and enlargers) with the ease of roll film.
Medium format uses roll film as well. The negatives are considerably larger: at 6 x 4.5cm, the smallest medium format, you get a negative that is a bit over three times as large as 135 (35mm). This presents the opportunity for high quality images (less enlargement to get the same size print with the same grain). The trade is bulk and convenience. The cameras are larger, the film yields less images per roll. Both the systems and the film is more expensive than small format.
Medium format includes point-and-shoot (the Holga), rangefinders, and SLRs. Medium has been the traditional choice for many commercial photographers.
Large format, at 4 x 5 inch and larger, affords the greatest image quality. With the ability to manipulate both the film and the lens plane, large format can simply do what other formats (generally) cannot. These cameras give the greatest amount of image control.
This comes at a cost of course, both literally and figuratively. The cameras themselves are large and bulky. They require a tripod (although there is at least one hand-held out). The film is loaded into holders, one or two sheets at a time, so either a light-tight changing bag or multiple holders are required. This along with a meter, a dark cloth...
Large format cameras come in a few general categories. First is the field camera (Wista, Toyo) which are designed to fold up into a more compact form. These cameras have the same kind of ability to manipulate the image, but the range of movements is limited. Field cameras are often largely made of wood, but contemporary ones may be constructed of metal.
The second type is the view or monorail camera. While certainly not limited to the studio, these cameras do not fold down compactly. They are, however, the ultimate in image control. Most are modular, with additional parts added as necessary. The basic parts are the rail, the rail clamp, the lens or front standard, the film or rear standard, and a bellows. Both the front and rear standard can be raised, lowered, swung side to side, and tilted forward and back, allowing for image control.
Also in this category is the press camera, such as the Graflex, which has limited movements, but is made to be portable. The cameras were in common use up until the 1970's. Many pinhole (lensless) cameras use large format film..