The wider the apature the more light is let in to the film. Therefor you need a faster shutter to limit the amount of light so you don't over expose the film.
Each f-stop you go down cuts the amount of light by 1/2 therefor you have to leave the shutter open twice as long for the same exposure.
A wide open apature will give you a shallow DOF (Depth of field). A very narrow apature will give you a long DOF.
Say you have a person on the street you want to shoot but you don't want to loose that person in the background of all the other stuff going on in the street. You would use a wide apature, say 2.8f so that only that person is in focus and everything else is blurred.
Or, you are taking a picture in a mountain meadow full of flowers with some great snow peaks behind it. You would use a small apature, 22f so that the flowers 2 feet away are in focus as well as the mountians several miles away.
Hope that makes sense.