Lenses that stop down nice and low are often referred to as “fast” lenses (or fast glass) and that is simply because they allow you to increase your shutter speed, because you are able to in more light via your aperture.
A lot of the time, especially when dealing with subjects that move – like children – it can be really helpful to work with a lens that has an aperture of F2.0 or larger. This will generally be a prime lens, since zoom lenses only stop down to around F2.8. (Some zooms also have a variable aperture, which means often they don’t even stop down to their maximum aperture when zoomed all the way in)
So, that is what a fast lens is, and why you might want one. But what exactly do you gain?
Well, let’s look at a "real life" example. Say you were shooting indoors using a zoom lens and it is quite an overcast day. Using the maximum available aperture of the zoom lens, your settings might look something like this:
F2.8 / ISO 3200 / 1/60
That’s probably too slow a shutter speed for photographing a person (and even too slow for some people to hand hold without a tripod) so you are probably going to have to increase your ISO by at least another stop to get 1/125 of a shutter speed, leaving you with an ISO6400. Using this high ISO's will lead to grain or noise in your images: how much depends on the camera model you are using - a full frame camera will handle this much better than a cropped body.
So, now we have these settings.
F2.8 / ISO 6400 / 1/125
But let's swap out that zoom lens for a prime lens, and in that exact same situation you would be looking at being able to use these settings instead:
F1.4 / ISO1600 / 1/125
What a much better ISO number! Still high-ish but will give you perfectly usable images after a little noise reduction.
If you need an even faster shutter speed, you could up your ISO to 3200 and get 1/250, or go all the way up to your original ISO of 6400, and get a shutter speed of 1/500. Hence the term "fast glass"!
That’s the main benefit of a fast lens, the ability to let in more light via your aperture, which in turn allows you to either increase your shutter speed, or reduce your ISO to keep noise at a lower level. Basically, it gives you more options. I don't own a zoom anymore, because I shoot indoors a lot (and in winter it can be really quite gloomy inside!) and I prefer to have that extra aperture range that a prime lens give me.
Of course, you could just get yourself a flash instead (and here's some examples so you can see why you might want one) but one of the other benefits of a prime lens is that you can get a very shallow depth of field, which does help your subject "pop" off the background by blurring the background effectively.
If you currently only have a zoom and find yourself regularly shooting at higher ISO’s and getting grainy images, or struggling to get a fast enough shutter speed, you might want to think about getting a fast lens too!