It is falling. It has to fall, that is the law of gravity. Otherwise, it would leave Earth in a straight line. But if it falls at exactly the rate it needs to fall to travel in an orbit, we call that "orbiting". If if falls faster than that, it crashes into Earth. If if falls less than that , it travels into space.
http://www.physicsclassroom.com/class/circles/u6l4b.cfm
The only force acting on a satellite in its orbit around Earth is the force of gravity from Earth. Why does a satellite not fall down?
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