Amateur (ham) radio is a hobby where people use radios to communicate with each other. An international agreement between the countries of the world has set aside a portion of the radio spectrum for private citizens to use and experiment with. You can transmit on these frequencies after passing a short licensing test. This is different from commercial broadcasting stations, in which both the equipment and the licensing is prohibitively expensive for individuals, and citizens' band (CB) or other unlicensed services which do not require a license at all to operate in. In return for required licensing and some restrictions, amateur radio allows you to do many things that CB or other unlicensed services do not.
Amateur radio operators (called hams) operate radio stations and experiment with wireless technology. There are many things you can do with amateur radio. You can remotely control model vehicles, spacecraft, or high-altitude balloons. You can run your own weather station and share the data with other people. You can track vehicles or runners in a race. You can send text messages or emails to people over radio without having a working cell or internet connection. You can talk to people around the world or even on the International Space Station. You can relay your radio signal through a satellite, or reflect it off the moon, or even bounce it off meteor showers and the aurora. You can design, build, and talk to other people using equipment you made.