Aperture - Photography
Aperture is measured using focal length. Also known as f-stop.
The lower the f-stop, the larger the aperture. The larger the f-stop, the smaller the aperture.
Aperture also controls depth of field. The smaller the aperture, the longer the depth of field. The larger the aperture, the shorter the depth of field.
If you have two objects at separate distances, regardless of where you focus, if you have a large aperture, one of the items may be out of focus. If you have a small aperture, they both may be in focus.
Now the best reason to get a lens with a large aperture (lower f-stop) is to let in more light. This in turn allows you to take shots at higher shutter speeds. You can always close up the aperture more, but you can not open up the aperture beyond what the lens says.
With that being said, you can understand aperture in this manner:
50mm f/1.8 50mm f/1.4
An aperture of f/1.4 means you can open up the aperture to the maximum of f/1.4. You can always close the aperture up more but you can not open up the aperture more than what is defined.
So between the above two lens definitions, f/1.4 will allow more light in than f/1.8. As such in lower light conditions you will want a lens which has a lower f-stop. Although professionally speaking, both are pretty damn good. There is a huge difference from f/1.8 to f/3.5 but not from f/1.4 to f/1.8.
A term people use to describe a lens with a low f-stop or large aperture is "fast lens". They essentially mean that you can take a picture at a faster shutter speed. The shutter is actually part of the camera so "fast lens" may not seem technically correct but as far as what is achieved, it's correct enough.
As a final mention, you can still shoot at a fast shutter speed even with an insanely high f-stop or small aperture. You most likely won't let enough light in to adequately expose the sensor / film, but I felt that it needed to be said. Just so you don't think you "can't" do it, you will just need more time to get a good exposure / shot. So more time basically means slower, less time means faster. Hence again, fast lens, lower f-stop (i.e. larger aperture).