Free climbing is the umbrella term for climbing styles where the rope and protection are used only to catch falls, not to make upward progress. The climber moves upward by pulling, pushing, and standing on natural rock features alone. Most modern climbing — bouldering, sport climbing, trad climbing, and the majority of multi-pitch routes — is free climbing.
The term exists in contrast to "aid climbing," where the climber pulls on, stands in, or hangs from gear (cams, ladders, hooks) to make progress on terrain too difficult or featureless to free climb. Aid climbing is still practiced on the biggest walls (Yosemite's El Capitan, Patagonia, the Karakoram) where parts of routes remain too hard to free climb cleanly.
Free climbing is often confused with free solo climbing — they're different things. Free solo climbing is climbing without a rope at all (made famous by Alex Honnold in the film Free Solo). Free climbing simply means not aid climbing — a climber on a sport route clipping bolts is free climbing. The rope catches falls; it doesn't make progress. Almost everyone reading this who climbs at a gym is free climbing every time they tie in.