Cars use a manual steering system or a power steering system. Manual steering systems do not have a power steering pump. When you turn the steering wheel, you directly turn the wheels.
With a power steering system, the power steering pump assists in turning the wheels. If the power steering pump system fails, you can turn the wheels, but it is hard to do so.
A belt turns the power steering pump. The power steering pump sends power steering fluid to the rack and pinion, or the gear box. If the belt breaks, or the pump does not work, it becomes hard to turn the steering wheel.
In most cases, the belt which turns the power steering pump also turns other devices such as the alternator, water pump, a/c compressor etc. A belt fails for two reasons. One, it can break. Two it can fall off the pulleys. If either of these two things happen, the power steering system stops working. As I said, you can still steer your car, but it becomes hard to turn the steering wheel.
If there is a leak in the power steering system, once you run out of power steering fluid, the power steering system stops working. You can still steer your car, but as I said, it becomes hard to turn the steering wheel.