A propeller is a marvel of engineering that, with the exception of the invention of the internal combustion engine, is probably the most important breakthrough in allowing us to accomplish powered flight. Yet a propeller is nothing more than wings rotating around a hub. If you look at the cross section of a propeller, you will notice it is shaped like a wing. On a wing the bottom side is flat, and the top is curved with the trailing edge tapered thin and the leading edge thicker. As a wing moves forward the flow of air is divided into the air that flows over the wing, and the air that flows under. The top side of a wing is curved, thus the air traveling over the top of the wing must travel farther to reach the rear of the wing. As the airflow over the top is traveling farther than the bottom airflow in the same amount of time, it must be traveling faster.
Bernoulli's Principle says that a rise in speed of a fluid, in our case air, occurs simultaneously with a drop in pressure, causing a low pressure in the fast moving air on the top of the wing. With a lower pressure on the top relative to the bottom, lift is created and the wing rises to try to equalize the pressure difference.
Now take two or more wings and attach them to a hub, rotate the hub, and you have a propeller. The rear side of a propeller is flat, the front surface curved. As the propeller rotates, the airflow over the front of the blade travels further relative to the airflow over the back, creating a low pressure in front of the propeller, and Bernoulli's Principle pulls the aircraft ahead.
A propeller also has some twist built into it, due to the fact that the tips of a propeller are traveling much faster than near the hub. The formula c=pi*d, or the circumference equals pi times the diameter. Because the diameter at the tips is double the diameter halfway out, so is the distance it must travel, and correspondingly its speed.
You will notice while looking at propeller tips from the side, the blades are nearly straight up and down, because of the high speed of the tips need less of a bite on the oncoming air to generate thrust. Closer to the hub more of a bite is required to generate thrust due to a reduced velocity, so the blade is gradually twisted to achieve the thrust more evenly over the length of the blade.
I'm sure many have heard the loud crack of a Harvard's propeller at an air show. When this happens the propeller tips are actually traveling supersonic due to blade length, and what you are hearing is sonic booms. On higher performance aircraft, designers will add more and shorter blades to a propeller rather than longer blades, to utilize the engine power, as supersonic blade tips are not only noisy but lose their thrusting efficiency.
Propellers are simply twisted wings, attached to a rotating hub, simple in concept but highly specialized to most efficiently turn engine power into thrust.