Lead Climbing
This form of climbing I'm not really qualified to speak on in depth, as I have personally never climbed this way before, but I'll give you a general idea of what it's like.
Lead climbing is much like top rope climbing in that you have a belayer and a climber. However with lead climbing, the climber takes the rope up the wall as they climb, clipping it in, so that if they fall, they fall from where they clipped it into. The belayer also has to potentially take and give rope, as the climber will need some leeway when clipping in.
Lead climbing is also often used when climbing outside, so if you have ever seen a photo of someone hundreds of feet of the ground, on the side of a mountain, chances are they might have been lead climbing.