Amateur radio, often called HAM radio, is a communication and electronics hobby regulated by the Federal Communications Commission. Specific frequencies are set aside for the exclusive use of licensees, providing communications from across town to around the world.
If you think of amateur radio as similar to Citizens Band Radio, there is a world of difference. CB is an unlicensed service with a limited coverage area for users.
Amateur radio is quite a bit more than voice communications. Amateurs have satellites in space giving them the ability to communicate with other continents, use slow scan television, link repeater stations to the Internet, and much more. Recent development of new modes of digital communication is helping bring new people into the hobby. Amateurs also use their equipment to provide disaster communications.
There are a number of sites providing insight into amateur radio. The American Radio Relay League is one, starting here.
Another good source is the online encyclopedia, Wikipedia.