Ysap and Matt gave great answers. I understand things best with metaphors, so here goes.

Consider a container that you wish to fill with water. In this case, the water will represent light, and the container will represent your lens. The wider the opening in the container (the aperture), then the faster you'll be able to fill the container with water (light).

If you were to take two containers, one with a very large opening and another with a very small opening, and dump the same amount of water onto each of them (for the same amount of time), which container would be filled with more water? Clearly the one with the bigger opening. If you wanted both containers to end up with the same amount of water, then the one with the smaller opening would have to have water dumped on it for a longer period of time than the one with the larger hole.

It works the same way with lenses. A camera with a small F-stop is a lens that can open up very wide, thus having a bigger opening to swallow up light. What this means to a photographer is that he/she can use much faster shutter speeds and still get adequate light in his or her photo. In this case, consider a swimmer coming up very briefly for a breath of air. The swimmer would much rather open his mouth up wide to take in air than purse his lips and try to breath in. A fast shutter speed with a wide aperture means more light can get in during the short period of time that the shutter is open.

Consider shooting a ballet in a dark theater. It would be best to use a lens with a wide aperture, because of the dark environment. Additionally, you would be able to use a faster shutter speed to capture the fast-moving dancers.

I hope this makes sense!