Planet orbits: what's the difference between gravity and centripetal force? My physics teacher says that centripetal force is caused by gravity. I'm not entirely sure how this works? How can force cause another in space (ie where there's nothing). 
My astronomy teacher says that gravity is (note: not like) a 3D blanket and when you put mass on it, the mass causes a dip/dent in the blanket and so if you put another object with less mass it will roll down the dip onto the bigger mass. Is this true and is this what causes the centripetal force. 
 A: Simple answer: gravity is a centripetal force, and can be envisaged clearly as such in Newtonian mechanics.
Centripetal just means a force that is "radially inwards" ("directed towards the centre"). The electric force between two objects of opposite charges, for example, is also clearly centripetal. (It's slightly harder to define "centripetal" for the magnetic force.)
Your astronomy teacher is referring to Einstein's theory of general relativity. His description is loosely an overview  of the topology (fabric) of space-time and how it interacts with matter/energy - the manifold is however 4-dimensional, not 3D.
In fact, test particles (particles which do not really disturb the gravitational field) in general relativity follow a geodesic. This is effectively a generalisation of a straight line (shortest route) of normal Euclidian space to the curved space of GR, and may be seen as the source of centripetal force in Newtonian physics.
A: A centripetal force is a force directed towards the centre. It's just a characterisation of an existing force. "Centripetal" means "towards the centre" in Greek.
So, in the solar system, the sun exerts a gravitational force towards itself, and it is a centripetal force.
Regarding your other question: how does gravity work? According to general relativity, energetic or massive objects distort space, so that other objects passing through the distorted space do not go straight, but bend their trajectory. From their point of view, they experience a force (gravity), and consequentially an acceleration which changes their trajectory.
So according to general relativity the medium through which gravity acts is actually the distortion of space.
This is what your astronomy teacher called "a 3D blanket".
Instead, according to Quantum Field Theory, forces are mediated by appropriate particles actually moving through space. Gravity would be carried by particles known as gravitons.
In both these theories, there is no action-at-a-distance, so body don't exert forces on one another instantaneously, but there's always something "in the middle" that carries or represents the force.
A: cetripital force only exists when you have prescribed motion (due to constraints). Think of a roller coaster car riding on a rail. To keep the car on the rail and tangetial to its direction a force and moment need to be applied to the car. When the path is circular we call the cetripetal force. In fact, with any path, instanteously it is said to be following a circle and therefore there is always an instanteneous centripetal force (unless in free fall).
Planets and things in orbit do not have a prescribed motion, but are following the free fall path whichever way they need to go. Gauss called this the principle of least action.
I hope this helps.
A: I think a key element that this student is missing is actually a very common misconception about centripetal force.  We hear of various types of forces, like normal forces, frictional forces, and gravitational forces, and then we tack on "centripetal force" as if it was another type of force like that.  But it's not-- those other forces are real forces with their own force laws and their own behaviors.  Centripetal force is not so much a name for a force at is a name for the mass times acceleration of an object moving in a circle.  So, centripetal force is not a kind of force, it is a net force that other forces have to add up to-- if and only if you already know the acceleration is that of an object moving in a circle.  So for an orbiting body with only gravity on it, gravity must be the centripetal force, but if a body is orbiting with both gravity on it, and a stretched rubber band going around as well, then the force of gravity plus the force of the rubber band will be the centripetal force.  In neither case is the centripetal force a type of force of its own.  (Oops, just noticed the date on the question, perhaps this will be of help to someone else.)
A: Gravity is a force.
Gravity is directed towards the center of the orbit i.e. the sun. 
That makes gravity the centripetal force. 
Imagine a ball attached to a string and you are holding the other end of the string and moving your hand in such a way that the ball is in circular motion. Then tension in the string is centripetal force. 
Now, 
ball = earth 
you = sun
tension in the string = gravity
Hope it helps. I have no idea about general relativity. 
