Film Formats
The smallest common film format is APS (Advanced Photo System). It has been developed by Kodak some years ago in an(other) effort to renew the low-end consumer market. The negative size is 16x30mm, and the closed cartridges contain up to 40 photos. The film also has a magnetic layer to store certain image information. The whole format is designed for print film only, and the very small negative size leads to problems even in 8x10" prints. Thus, this format is to avoid for the serious photographer. The film choice is very small, the format is too small, it's difficult to develop the films on your own, and we don't know whether it has a future at all. There is also no option to shoot APS chromes. Actually it would be much cheaper to use half-frame cameras using regular 35mm film if one wants to have a smaller negative for economical reasons.
The by far most common film format is 35mm (German "Kleinbild" - small frame). These are cassettes containing 36 (1.65m long) or 24 photos sized 24x36mm each. This format has been invented by Oskar Barnack in the 1920s using movie film. Today, most professional photographers use it, and it delivers prints of up to 20x24" in acceptable quality if the photo is carefully achieved. Note that 35mm was known as "double frame" in the early days, as it used an image area twice as large as the 18x24mm used for movies. There are certain photo cameras which produce a negative sized 18x24mm on 35mm film (thus 72 photos instead of 36) which is known as "half frame" today.
Medium format actually comprises several different formats. According to the German industry norms (DIN), all films with a negative size larger than 35mm (24x36mm) and smaller than 9x12 (90x120mm) are medium format (by the way, this norm also implies that the panoramic roll film format 6x17 is still medium format). But in fact, today all formats using roll film (type 120 is 80cm long, type 220 is twice that) are commonly understood as medium format. The most common ones are 4.5x6 (42x56mm; 16 photos per type 120 roll), 6x6 (56x56mm; 12 photos), and 6x7 (56x70mm; 10 photos). There are many older cameras using 6x9 (56x83mm; 8 photos). Medium format doesn't usually have autofocus, but the bigger negative delivers much higher quality. As a general rule, you can say that a low-end medium format camera will always outperform a high-end 35mm camera, simply due to the larger film size. Wedding, studio and landscape photography are classic areas of medium format.
Large format also consists of many different formats. It uses sheet film sized 9x12 (90x120mm) or larger. The largest common one is 8x10" (203x254mm), but ultra large format goes up to 20x24" (508x610mm). Probably the most common large format is 4x5" (102x127mm). Large format doesn't offer any automatic aids to the photographer, but its large negative is very useful for landscape photography. In addition, most large format cameras offer many features for perspective and depth-of-field control which are also often necessary in architectural, product, and landscape photography. As far as the film choice is concerned, large format is the most flexible format, because you cannot only use sheet film and Polaroids, but there are roll film and 35mm backs as well.