Elaborating on tmarthal and tmac's answers:
The force which acts on a body undergoing circular motion, always pointing towards the centre of the circle. This force ensures that the body remains at a fixed distance from the center of the circle. It can be gravitational, electrostatic, magnetic etc. Edit: It can also be a combination of two or more different kinds of forces.
Think of it this way:
The green thingy is a ball whose velocity is initially along the black horizontal line. Now I apply a force along the red line, which is perpendicular to the black one.
This causes the ball to turn ever so slightly downwards, so that it has a new velocity (v').
At the same time, I adjust the direction of my force so that I have a new force (F') which is perpendicular to v'.
So the velocity keeps on changing direction, and I keep on adjusting the direction of my force so that the ball moves in a circle. At any point, the force is perpendicular to the velocity.
Notice that had there been no force, the ball would simply have moved in a horizontal straight line. The force 'forces' the ball to move in a circle. ;-) No force is required to 'balance' the centripetal force.
Also, in this example I would have to continuously change the direction in which I apply a force on the ball, so as to keep it moving in a circle. In your example, the gravitational force on the satellite is the centripetal force which keeps the satellite moving in a circle. It is always acting towards the center of the earth, so the 'adjusting' happens automatically as the satellite moves around the earth.
This one is a little more complicated. It's best explained through an example.
Imagine that you're sitting on a merry-go-round which is spinning in a circle. You are moving in a circle on the merry-go-round. Hence, there must be some force which is acting on you, pulling you towards the center of the merry-go-round. In the absence of such a force, you would fly off the merry-go-round. It's easy to guess that here, the centripetal force acting on you is composed of (1) the friction between you and the floor of the merry-go-round and (2) the force applied by the handlebars and other parts of the thing on you, as a result of you pushing against them.
Now this is the important part - you, sitting on the merry-go-round, believe that you are not accelerating. (A person standing on the ground knows that you are accelerating, as your velocity is continuously changing its direction thanks to the centripetal force acting on you). Since you believe that you are not accelerating, you believe that the net force acting on you is zero. Thus, you feel an outward force acting on you, which 'balances' the centripetal force provided by the handlebars and friction. This is the so-called centrifugal force.
This is true for all accelerating frames - anyone inside the frame thinks that they are not accelerating, and hence feels the effect of a "pseudo" force which acts opposite to whatever real force is acting on them and making them accelerate.
To summarize, centripetal force is a real force which must act on a body for the body to travel in a circle. It acts towards the center of the circle. Centrifugal force is a force felt by a body if it is sitting in a frame which is travelling in a circle. It acts away from the center of the circle.