Dave's answer is correct, but I have a more graphic way that I like to explain it.

Look at the image below. Now, for this example, let's say that the wing is symmetrical, so there's little or no lift involved in the process.

When the wing is banked, the plane still wants to fly straight. However, as gravity pulls the plane down, it will want to slide in the direction of Arrow "A".

When backpressure is applied to the elevator stick, the elevator will cause the nose to go in the direction of arrow "B".

The combination of "A" + "B" will bring the plane in direction "C".

So it's really a balancing act, too much elevator will cause the plane to climb through the turn, while too little will cause the plane to dive.

Now, with a flat bottom wing, we add lift to the equation. As the plane turns, the outer wing is moving through the air faster than the inside wing, therefore, it is generating more lift. The dihedral gives the wing a lot of self-righting characteristics, but you CAN get into a situation where the turn becomes so tight that the plane will want to flip over due to the excessive speed/lift of the outside wing. This is something to be careful of when hot-rodding with a trainer type plane.

Now let's look at a Rudder turn.

Typically, a 3 channel plane will have a flat bottom wing, so as the rudder is applied, the outside wing moves faster causing the plane to bank, but as the plane banks, the rudder is now turning the nose toward the ground. Up elevator helps, but only to a point, and can likely cause the plane to go into a spin. Ok, so you manage to avoid a spin, but now the plane is banked, and you want to correct the turn, so you apply opposite rudder. Since the plane is banked, the rudder also now acts to RAISE the nose, causing a Ballooning effect when you pull out of a turn.

There is some controversy as to whether or not the Wright Bros were the first to fly. It really doesn't matter, what the Wright Bros WERE the first to do, was to fly a plane that could BANK through a turn! We still use a version of their idea today.