Honda made the first ATC models for use by farmers. The ability to go anywhere on terrain that most other vehicles could not cross soon made them popular with US and Canadian hunters, and those just looking for a good trail ride. Soon other manufacturers introduced their own models.
Safety issues with 3 wheels caused all manufaturers to switch to 4-wheeled models in the late 80's, and 3-wheel models are no longer made. The lighter weight of the 3-wheel models make them popular with some expert riders. The safety issues were in cornering: all models have a solid rear axle, so to corner the rider needed to reduce speed more than expected, and to lean out. This is contrary to ones instincts and to the behavior of all other vehicles, so many riders were injured.
Models today are roughly divided into the sport and utility markets. Sport models are generally small, light, 2 wheel drive, and go fast. Utility models are generally bigger, 4 wheel drive, max out at about 45mph, and have the ability to haul small loads. Due to the different weights, each has advantages on different types of terrain.
6 wheel models have a small dump bed, with an extra set of wheels at the back to increase the payload capacity. They can be either 4 wheel drive (back wheels driving only), or 6 wheel drive.
There also exist 6 and 8 wheel models where the rider sits inside. These vehicles float and are designed to go though swamps as well as dry land.