Satellites are, to some degree, "mysterious" objects. They travel in space, which feels like an exotic place because most of us have never been there. They are so far away that we cannot see them. They usually cost millions or billions of dollars, which means none of us will ever own one personally. And so on…
Orbital mechanics can also be mysterious because there is no easy way for us to experience orbital mechanics personally. However, with a little imagination, you can understand the basic idea behind orbital mechanics very easily. It turns out that we play with orbital mechanics all the time!
Think about what happens when you throw a ball. Imagine that you are standing in a big field and throw a baseball as hard as you can -- like a pitcher. The ball might go 100 feet (30 meters) and then hit the ground. You put the ball in orbit -- It's just that a ball's orbit is very short!
Now imagine that you shot a rifle straight and level instead of throwing a ball. The bullet might travel a mile (1.6 km) before succumbing to gravity and hitting the ground.
Now imagine that you shoot a very large cannon that is able to give its shell an extremely high initial velocity. Also imagine that our world is completely covered in water to remove any worries about hills, and that the cannon is shot straight and level. Its path might look like the image to the right.
In this diagram you can see that the shell is going far enough to actually follow the curve of the earth for a period of time before hitting the ground.
One thing that gums these examples up is air resistance, so imagine that you took this cannon to the moon and mounted it on top of the highest mountain. The moon has no atmosphere and is completely surrounded by the vacuum of space. If you adjusted the speed of the shell just right and shot the cannon, the shell would follow the curve of the moon perfectly. It would fall at exactly the same rate that the curve of the moon falls away from it, so it would never hit the ground. Eventually it would curve all the way around the moon and ram right into the back of the cannon! On the moon you could actually have satellites in extremely low orbits like that -- just a mile or two off the ground to avoid the mountains. And satellites could conceivably be launched from cannons.
On earth, it's not so easy because satellites have to get up above the atmosphere and into the vacuum of space to orbit for any length of time. 200 miles (320 km) up is about the minimum to avoid atmospheric interference. The Hubble space telescope orbits at an altitude of 380 miles (600 km) or so. But the principle is exactly the same. The speed of the satellite is adjusted so that it falls to earth at the same rate that the curve of the earth falls away from the satellite. The satellite is perpetually falling, but it never hits the ground!
Here are several interesting links: