The force of gravity, which defines the motion of the satellite, is proportional to mass, so at a particular distance the acceleration due to gravity will be the same on different objects.
Consider an object in open space with no forces acting on it moving with constant velocity. When that object enters a gravitational field the acceleration due to gravity will act on the body. Three things may happen.
The object is travelling wide of the centre of the gravitational field, it is deflected by the field but has sufficient momentum to escape the field. It has been deflected by the field, it will curve around the field and continue on into space.
The object is travelling nearer to the centre of the gravitational field. As it is deflected by the field it moves closer to the centre of the field and the acceleration due to gravity increases. It does not have suffient momentum to escape and ends up crashing into the planet or star that is generating the field.
The object does not have enough momentum to escape the planet, but has just enough momentum to be captured by the planet. In this case the force of gravity keeps the object moving around the planet. The planet will be at one of the focii of an elipse that describes the orbit of the object.
We expect, and observe, that 3 is a fringe case. Most objects either pass on or end up crashing into the planet or star. Objects which are captured remain in orbit.
For man made satellites we can plan the orbit and as such calculate how to put the satellite into that orbit. There are lots of possible configurations for 'stable' orbits which may be geo-stationary or not. As we ubnderstand what the force of gravity is, and therefore the acceleration due to gravity, we can plot an orbit and calculate what the tangential velocity of a satellite at any point in that orbit would be. We can then put a satellite into the correct position, with the correct tangential velocity, and just 'let it go'.
There's a good explanation of the mathematics of orbital motion at this Wikipedia page.
