The aperture is measured in a very funny way that is somewhat hard to understand.
The aperture is defined with respect to the focal length of the lens:
A = f/d
- A is the aperture value
- f is the focal length of the lens
- d is the diameter of the opening of the lens
Let’s look at an example.
Take a lens with a focal length of 3 inches that is 1 inch across. The aperture is
Aperture = f/d
= 3 / 1
In optics and in photography, this is properly written as,
That is, the aperture is about one-third of the focal length of the lens. This emphasizes the relationship between the aperture and the focal length.
Thus, if the lens (that is, the opening in the lens) is smaller, the aperture value gets smaller, which is a larger number in the denominator.
Notice that if you convert everything to millimeters, you’ll get the same value for the aperture—it is a ratio.
With modern lenses, the relationship between focal length and lens opening is a lot more complicated. But it is a good start to say that the aperture value is the focal length of the lens divided by the opening in the lens.
A bigger aperture (smaller number in the denominator of the aperture fraction) lets more light hit the sensor; A smaller aperture (a larger number in the denominator of the aperture fraction) lets in less light.
Here is a good way to understand how a bigger lens/larger aperture lets in more light. Get a magnifying glass and go out on a sunny day. Now set fire to a piece of paper by focusing the sun on it.
Now get a piece of cardboard and put a generous hole in it—about 1/3 of the size of the lens. Obscure most of the lens with the cardboard, but let the central part of the glass focus the light through the cardboard onto the paper. You can still burn the paper, but it takes a lot longer. This is because less light is being allowed to hit the paper.
You get the same effect with a large lens and a small lens. Try it.
 This section is here for completeness. It can be skipped.
 The focal length of a single convex piece of glass is described succinctly in the Wikipedia: http://en.wikipedia.org/wiki/Lens_%28optics%29#Lensmaker.27s_equation . A full explanation of focal length is deferred to another manuscript.
Next: Aperture values