A dream becomes lucid when you consciously realize that you're having a dream. It's that 'a-ha' moment you get when you finally realize that talking to a half-man, half-cat hybrid isn't exactly normal.
In this state of awareness, it's also possible to learn to control what's happening in the dream. You can learn to decide the location, the people, the events taking place, literally anything you can imagine becomes possible.
Want to fly through the air shooting fireballs out of your hands? Go ahead, the only thing stopping you is the belief that gravity still exists!
While being a ton of fun, lucid dreams are quite often a rare occurrence for most. While some people have them on accident, with the right training it is possible to purposefully induce lucid dreams.
You can develop 'reality checks' that help you consistently determine when you're dreaming. By doing this you can eventually learn to stay in the dream and alter its course.