A satellite is a celestial body that orbits another celestial body. This could be a moon(satellite) around a planet (earth), or a planet(satellite) around a star.
(moon around earth, earth around sun) - examples
When an object orbits around a gravitational mass such as the earth, it is pulled towards the center of the earth by the centripetal force provided by its weight. If the object is moving quickly enough, the Earths surface will curve away from it by a equal distance to its fall, hence the object effectively stays in its orbit. This state is known as freefall.
The formula for eccentricity is:
$$ e = \sqrt{1+\frac{b^2}{a^2}} $$
Where $b$ is the semi-minor axis and $a$ is the semi-major axis.
A satellite which orbits around Earth in a geosynchronous orbit has the same orbital period as Earth. Geosynchronous refers specifically to this kind of orbit around Earth. Similarly, the Areosynchronous orbit is for Mars.
A geostationary orbit is a special case of a geosynchronous orbit. It is where the satellite is orbiting exactly on the equatorial plane.