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 (26,200 miles).
In the special case of the geostationary orbit, an observer on the ground would not perceive the satellite as moving and would see it as a fixed point in the sky.
Such orbits are useful for telecommunications relays.
In the more general case, when the orbit has some inclination and/or eccentricity, the satellite would appear to describe a more or less distorted figure-eight in the sky, and would rest above the same spots of the Earth's surface once per sidereal day.
Synchronous orbits exist around all moons, planets, stars and black holes —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.
See the following related content on ScienceDaily: