When you hear the term 'fast lens' it means that the lens in question has a large maximum aperture (the bigger the aperture, the faster the lens will be). The aperture is often displayed as an f followed by a number but do remember that a large maximum aperture will actually be a small number such as f/1.8. A fast prime lens would be considered fast when it has a maximum aperture under f/2.8. However, if the lens is 300mm or longer, an aperture of f/2.8 would be considered to be fast and the same goes for zoom lenses.
A bigger aperture (small f number) will allow more light to reach the camera's sensor which means faster shutter speeds can be used even in low light situations. They're useful in various shooting situations including places where flash can't be used, at concerts where there's not much ambient light, indoors when you're trying to capture movement such as dancers on stage and for subjects such as sports photography where fast shutter speeds are essential.
Another advantage to fast lenses is that you won't always be forced to use a tripod as the faster shutter speeds allow for hand-held shooting in more situations. This is something that's particularly useful in places where tripods aren't allowed such as in cathedrals or in busy locations where light can be an issue such as in a museum.
A downside to fast lenses is that they can be expensive and they tend to be heavier and bigger than other lenses. Care needs to be paid to focus when using autofocus as you may find it tries to focus on the wrong part of the shot, leaving focus on an area of the image that wasn't your intended subject. It's also worth investing in a good quality lens so images don't appear soft when viewed on-screen.