How does a space station stay in orbit?

Like any other satellite, including the Moon, it stays up by centripetal force. Like a weight being whirled around your head on a string it would fly off in straight line if the force holding it in were to disappear. With the weight, this force is the tug of the string. With the space station it's gravity. The speed of the space station is just sufficient to keep it from falling to Earth, but also not so great that it would fly off into space.

If the speed of the space station were to drop, it would fall to Earth.