A satellite (be it man made or a natural satellite such as the moon) stays in orbit due to the gravitational field of the planet.
At any particular height, there is one speed at which a satellite can maintain a permenant orbit. As gravity pulls the satellite down, the forward speed of the satellite is unaffected. This means that the satellite travels in a curved, elliptical path and will always follow the same path around the planet.
Satellites can be placed at different heights so that they circle the Earth over different time periods. A low orbit of a few hundred kilometres means that the satellite will circle the Earth in less than two hours. A satellite placed at a height of around 36,000km will take 24 hours to circle the Earth. If placed above the equator, it will maintain a geostationary orbit; it will always look to be in the same position above the Earth.