When the talk about a fast lens, it's a reference to the f-stop at wide-open aperture. A lens "faster" than f/2.8 would be one with a wider maximum aperture (lower f-stop). For example, Canon has 50mm lenses in both f/2.8 and f/1.4. The f/1.4 lens is "faster" because its maximum aperture is wider, meaning it will admit more light and therefore you can use a faster shutter speed at that f-sop to get the same exposure as with the wide-open f2.8 lens. At any f-stop above 2.8, however, all thihgs are equal for both lenses.

Hope this makes sense. I haven't had nearly enough coffee yet this morning....

Almost true. The image quality may be similar but the image will be a lot easier to see in the viewfinder and focusing will be a lot faster (both autofocus or manual focus) with faster lenses, because they do let in more light. One stop faster is twice as much light.

Larger apertures let in more light and have lower f-stop numbers at maximum aperture and are faster (allowing you to use a faster shutter speed) lenses. F 1.4 is faster than F 2.0 which is faster than F 2.8, etc. Faster lenses also cost a premium and are physically larger and heavier. But they can be worth their weight in gold if you're in a situation where you can't make the picture without one.