A geosynchronous orbit is a geocentric orbit that has the same orbital period as the sidereal rotation period of the Earth. It has a semi-major axis of 42,164 km  (http://www.google.com/search?q=%28%2886%2C164.09%2F2%2Fpi%29%5E2*398600%29%5E%281%2F3%29).
Synchronous orbits exist around all moons, planets, and stars —unless they rotate so slowly that the orbit would be outside their Hill sphere. Most inner moons of planets have synchronous rotation, so their synchronous orbits are, in practice, limited to their leading and trailing Lagrange points. Objects with chaotic rotations (such as Hyperion) are also problematic, as their synchronous orbits keep changing unpredictably.
If a geosynchronous orbit is circular and equatorial then it is also a geostationary orbit, and will maintain the same position relative to the Earth's surface. If one could see a satellite in geostationary orbit, it would appear to hover at the same point in the sky, i.e., not exhibit diurnal motion, while one would see the Sun, Moon, and stars traverse the heavens behind it.
A circular geosynchronous orbit in the plane of the Earth's equator has a radius of approximately 42,164 km (from the centre of the Earth) or approximately 35,790 km (22,240 statute miles) above mean sea level.
This can be demonstrated analytically by application of the Law of Gravity and the physics of centripetal acceleration. Drawing the free body diagram and using the analysis methods of engineering dynamics and physics allows the determination of the distance from Earth's centre of mass which will satisfy this specified operating condition.
Circular geosynchronous orbits
Circular geosynchronous orbits at the equator are known as geostationary orbits. A perfect stable geostationary orbit is an ideal that can only be approximated. In practice the satellite will drift out of this orbit (because of perturbations such as the solar wind, radiation pressure, and the gravitational effect of the Moon), and thrusters are used to maintain the orbit.
See Geostationary orbit.
Other geosynchronous orbits
Elliptical orbits can be and are designed for communications satellites that keep the satellite within view of its assigned ground stations or receivers. A satellite in an elliptical geosynchronous orbit will appear to oscillate in the sky from the viewpoint of a ground station, tracing an analemma in the sky. Satellites in highly elliptical orbits must be tracked by steerable ground stations.
Theoretically an active geosynchronous orbit can be maintained if forces other than gravity are also used to maintain the orbit, such as a solar sail. Such a statite can be geosynchronous in an orbit different (higher, lower, more or less elliptical, or some other path) from the conic section orbit formed by a gravitational body. Such devices are still theoretical.
A further form of geosynchronous orbit is obtained by the theoretical space elevator in which one end of the structure is tethered to the ground, maintaining a longer orbital period than by gravity alone if under tension.
Author Arthur C. Clarke is credited with popularizing the notion of using a geostationary orbit for communications satellites. The orbit is also known as the Clarke Orbit.
The first communications satellite placed in a geosynchronous orbit was Syncom 2, launched in 1963. Geosynchronous orbits have been in common use ever since, including satellite television.
Initially, geostationary satellites also carried telephone calls but are no longer used so predominantly for voice communication, partly due to the inherent disconcerting delay in getting information to the satellite and back (it takes light or radio about a quarter of a second to make the round trip).
Nearly all locations on the planet now have terrestrial communications facilities (microwave, fibre-optics), even undersea, with more than sufficient capacity. Satellite telephony is now mainly limited to small, isolated locations that have no terrestrial facilities, such as Canada's arctic islands, Antarctica, and the far reaches of Alaska and Greenland. These locations still use satellite telephony, and innovations have made it possible for internet transmissions to cope with the time delay.
- Nasa.gov: Geosynchronous Orbit (http://liftoff.msfc.nasa.gov/Academy/ROCKET_SCI/SATELLITES/geo-high.html)