Ham Radio contests have been likened to the worlds biggest video game. And you don't even need a computer to participate! The goal of most Ham Radio contests is to see how many stations you can contact in a given time period. To compete on an international level takes a great deal of skill and practice. It is much like a video game or many sports, you plan strategy, compete in real time with others around the world, and feel the thrill of victory. And even if you aren't competing on a high level there are catagories for the casual operator to get in and have fun, much like the fun runs and walking classes associated with marathons and other events.
Most contests are divided up into catagories by the size of the stations and the number of operators. This is similar to classes in car or boat racing. Every contest(I think) has a class for an individual operator working alone, and many have classes for 2 or more operators using one or more radios at the same time. In large events there are teams of a dozen or more that work together much like the crew of a sail boat on an open ocean race. And many events have club competitions where various sized clubs combine the scores of all their members to compete for awards.
Contests are sponsored by many different clubs, magazines, and national ham radio organizations like the American Radio Relay League. These organizations collect and validate scores from all the entrants and then publish results. There are many contests that are also sponsored by state or local organizations. Just about every weekend there is some kind of contest running somewhere. For a large, but hardly complete list see my general schedule of upcoming contests.
ZS1AN Contesting FAQ
K2YWE Contesting presentations