A harness is the connection from the climbing rope to your body. Harnesses have evolved tremendously over the last few decades, and are now lightweight and comfortable even when sitting in them for hours on end.
Sport climbers want to limit weight and maintain range-of-motion. A sport climber's harness has little to no padding and often only two gear loops. These harnesses are low profile so they don't impede the climber's movements, and they have pre-set leg loops to save weight. A sport harness isn't comfortable to hang in for long periods and has a very low rated haul loop.
All-around harnesses add thicker, and wider, foam padding at the expense of weight. These are a great choice for beginners or folks who want a harness that can do it all at a low price. Outfitted with four gear loops and many options for buckles/adjustability, it's easy to find a great harness that fits well and works in all situations.