When you ride a bicycle, you are cycling. Sometimes, though less often, it also is used to refer to motorcycle riding.
The roots of the word cycling are ancient. Both Latin and Greek have similar words that mean "circle, wheel." It's only natural, then, that a two-wheeled vehicle should be a bicycle and that cycling involves riding that vehicle. Occasionally, it can be used as a verb describing something going through a cycle, like a dishwasher cycling through.