It seems very counter intuitive that the amount of light let in by a lens is referred to as the lens speed. This term confused me for a while because I couldn't logically see how a speed would be associated with a particular lens, being either fast or slow.
Is there any reason why this is referred to as the speed of the lens?
Indeed, Speed is way too overloaded in Photography, we have lens speed, shutter-speed and ISO speed!
These terms indirectly refer the possible shutter-speed. A fast lens has a bigger maximum aperture (represented by smaller numbers) which lets more light in. As a consequence, a fast lens lets you use fast shutter-speeds.
The same is true of ISO. A high ISO is called fast because you can use a faster shutter-speed with it.
Consider the exposure triangle - aperture, shutter speed and ISO. When you adjust one, you can adjust the others accordingly. Ignore ISO for now, as that is a function of the camera, and consider the aperture and shutter speed variables.
The wider the aperture, the more light can enter the camera. This means that for any given exposure, the shutter speed can be shorter - in other words, the shutter speed is fast. Faster shutter speeds are generally desirable as they freeze motion and counteract camera shake.
So a fast lens is generally a wide aperture lens, like a 1.4 or 2.8, because compared to a slow lens like a 4 or 5.6, you can get many more shutter speed stops with them.
Consider that you are taking a photo with a specific amount of light, say a dark-cloudy day. You get an exposure setting from the ASA rating of the film or digital sensor, the f-stop and the shutter speed.
If you put a F5.6 lens on your camera body, you will have to have a slow shutter speed.
Put an F1.4 lens on it, and your shutter speed will be must faster.
As @ital says, we tend to overload the term speed in many areas within photography.
