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Within photography the general scale for things relative to each other is 'fast and slow'. This applies to shutter speed, film speed, and lens speed.

I understand shutter speed, and kind of have a grasp on film speed, but lens speed?

Yep, lenses are often talked about as being 'faster' or 'slower' than each other. This refers to the maximum aperture of the lens, so a f/1.4 50mm lens is faster than a f/2 50mm lens. Typically this comparison is made between lenses of the same focal length - it is 'silly' to compare a f/1.2 50mm lens to a f/5.6 300mm lens because the f/stop itself is a relationship between the focal length of a lens and the size of the aperture.

So, why would you want a fast lens? Aren't faster lenses more expensive too?

Two reasons mainly, and yes - they are more expensive, they have more glass in them. The first is the obvious one - with a faster lens, you can use a faster shutter speed. This is often critical with low light photography. Compare using a f/5.6 lens to a f/2.8 lens. The f/2.8 lens allows four times as much light in as the f/5.6 lens does. This means that the same film must be exposed four times longer and thus allowing for much more motion in the photograph. When taking stop action photographs (cars at a race track or sports games) the shutter speed should be 1/500s or faster. Granted, moving to a faster film can compensate for having a slower lens, however implies a loss of quality in the film. Think fast action is better with a fast lens.

The second reason for a fast lens has to do with depth of field. When taking pictures of a subject, typically the photographer wants to have the attention of the viewer on just that subject. It would be unfortunate for the photograph to have everything in focus from the subject to the background in focus. The most extreme form of this (with the slowest of all lenses) is that of the pinhole camera where everything is in focus (and very little light gets through). With a larger aperture (f/4 to f/1) only the subject is in focus and everything else is a blur. This is also useful in taking pictures of sports and racing (again) where it is just the athlete or the car that should be in focus, not the bandstand and crowd behind it.

In many point and shoot cameras, the lens aperture (most often) or shutter speed (rarely) is fixed. This then has the function of the sunny sixteen have only two variables: the film speed and the shutter speed and thus leads to the essence of the sunny sixteen rule:
shutter speed = film speed.

Case study:
You've got a 200 speed film loaded in a point and shoot camera (an old one) with a manual switch for what speed film is loaded: 100, 200 or 400. Today happens to be overcast. By changing the speed of the film the camera thinks that it has loaded to 100, the shutter speed will be changed to 1/100s from 1/200s.

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