Short Answer: It's a combination of inertia and gravity.
Long Answer:
In space, where there is (almost) no air resistance because there is (almost) no air, a moving object would tend to just drift along in a straight line at a constant speed, unless a force is applied that changes the direction of the object's movement (eg: Collision with another object, firing a rocket, etc). This tendency to keep "moving right along" is called inertia.
In the case of an orbiting satellite, the force being applied is the Earth's gravitational pull, which tries to pull the satellite back towards the Earth's surface. The further away from Earth you get, the less effect the force of gravity will have.
So, to put it all together, NASA (or whoever) release a satellite into space at just the right speed so it is not going quite fast enough to drift away and escape Earth's gravity, and at just the right distance from Earth so gravity can't quite pull it back down.
The combination of the satellite's inertia and the Earth's gravity results in the satellite travelling in a circular path around the Earth, ie: A stable orbit. It is exactly the same combination of forces that cause the moon to orbit the Earth, and the Earth and other planets to orbit the Sun.
Of course, there are some problems with this. Space isn't a perfect vacuum, so there's a tiny amount of "air resistance", causing ever-so-slight changes in the satellite's movement. Secondly, even NASA can't get the speed and distance accurate enough to keep the satellite in a stable orbit for much longer than a few years.
Without slight corrections to the satellite's course, one of two things happens: Either the satellite ends up travelling too fast and escapes Earth's gravity, floating away into space, or gravity eventually wins this little game of celestial tug-of-war, and the satellite comes back down too close to the Earth, entering the atmosphere and burning up.
To extend a satellite's lifetime, small rockets can be built into the satellite, and used to correct its orbit, or a satellite can be captured by a spacecraft (eg: Space Shuttle) and moved back into a more stable orbit. In most cases, though, a satellite will outlive its usefulness well before this needs to happen. SkyLab, for example, was going to be moved into a higher orbit by the first Space Shuttle in 1979, but when the first Space Shuttle's completion was delayed for two years, Skylab was left to re-enter the atmosphere and burn to pieces. The Hubble Telescope might be an exception to this - it will probably be useful for many years to come.
Hope this answers your question to your satisfaction.
-B
Long Answer:
In space, where there is (almost) no air resistance because there is (almost) no air, a moving object would tend to just drift along in a straight line at a constant speed, unless a force is applied that changes the direction of the object's movement (eg: Collision with another object, firing a rocket, etc). This tendency to keep "moving right along" is called inertia.
In the case of an orbiting satellite, the force being applied is the Earth's gravitational pull, which tries to pull the satellite back towards the Earth's surface. The further away from Earth you get, the less effect the force of gravity will have.
So, to put it all together, NASA (or whoever) release a satellite into space at just the right speed so it is not going quite fast enough to drift away and escape Earth's gravity, and at just the right distance from Earth so gravity can't quite pull it back down.
The combination of the satellite's inertia and the Earth's gravity results in the satellite travelling in a circular path around the Earth, ie: A stable orbit. It is exactly the same combination of forces that cause the moon to orbit the Earth, and the Earth and other planets to orbit the Sun.
Of course, there are some problems with this. Space isn't a perfect vacuum, so there's a tiny amount of "air resistance", causing ever-so-slight changes in the satellite's movement. Secondly, even NASA can't get the speed and distance accurate enough to keep the satellite in a stable orbit for much longer than a few years.
Without slight corrections to the satellite's course, one of two things happens: Either the satellite ends up travelling too fast and escapes Earth's gravity, floating away into space, or gravity eventually wins this little game of celestial tug-of-war, and the satellite comes back down too close to the Earth, entering the atmosphere and burning up.
To extend a satellite's lifetime, small rockets can be built into the satellite, and used to correct its orbit, or a satellite can be captured by a spacecraft (eg: Space Shuttle) and moved back into a more stable orbit. In most cases, though, a satellite will outlive its usefulness well before this needs to happen. SkyLab, for example, was going to be moved into a higher orbit by the first Space Shuttle in 1979, but when the first Space Shuttle's completion was delayed for two years, Skylab was left to re-enter the atmosphere and burn to pieces. The Hubble Telescope might be an exception to this - it will probably be useful for many years to come.
Hope this answers your question to your satisfaction.
-B