Along with Shutter Speed and ISO, the aperture is part of the Exposure triangle. It's the size of the hole in the lens that light passes through. Make the hole bigger, more light gets in. Double the size of the hole, but leave everything else the same, the image will get twice as bright. Double the hole, and halve the time the shutter is open, you get the same exposure, but still get a different image.
Beyond getting the exposure you want, the main effect of aperture is to get the Depth of Field you want. It's basically just how much of your image is in focus, and just how blurry those background bits are if you want that Bokeh.
- Want everything in focus, as you tend to see with Camera Phone shots, and most Street Photography? Narrow aperture, or big number. f/8 usually gets a lot in focus, but it depends on your focal length and how far away things are.
- Want the sort of portrait photo where the background is all blurry and the person seems to pop out of the image? Shallow depth of field, so you want a big aperture, or a small number. f/1.4 perhaps, if your lens can do it, or f/2.8. Again, depends on other things too.