If you have done much shooting, you've likely become familiar with how shutter speed and aperture go together. Change the shutter speed and you must adjust the aperture the opposite direction to keep the total light the same. The aperture is the 'hole' in the lens that adjusts how much light actually makes it through. The aperture can be adjusted from completely open, to almost closed (no point in actually closing it!). The aperture is usually anywhere from 5 to 8 blades, sometimes curved. Milosluz's shot here shows an aperture with 7 blades. While it is very difficult to actually get a perfect circle when the aperture is closed down, it is pretty close for the purpose of coming up with the F number.
So far so good. But let's take a look at what happens when you change the aperture. Let's say you 'stop down' to a 5mm hole in the lens. This reduces the light coming in, and your F number is now 5.6 (28mm/5mm). The first thing to notice is that the F number got 'bigger'. From 2.8 to 5.6. Yet the hole went smaller, and lets in less light. This is mostly a matter of convenience. They could have defined things the other way, and flipped the equation over, in which case we would have gone from 1/2.8 to 1/5.6 - and it would make more sense. This is much more easily seen as making things smaller. But it is cumbersome to always talk in fractions, and takes a lot more space on the aperture ring. This is just something you need to understand and get used to. (It is the same with shutter speeds, by the way. Modern cameras show just half the story - 250 when they really mean 1/250s).
To make things more complicated, most aperture rings allow half stops between the F numbers. Camera bodies these days allow for third stop settings. So a 'click' on the dial is not usually a stop.
Here are a few other things about apertures:
1. From the above you can see why it is hard to make a local focal length lens with a wide aperture. For a 135mm lens to have an aperture of f/2, it needs a hole around 67mm. And Canon's 85mm f/1.2 needs one about 70mm - hence the large amount of glass on the end of the barrel. Note that we are talking about the effective aperture, so there isn't necessarily as simple as this. People make 400mm lenses with an aperture of f/2.8, without needing a 140mm filter :) Conversely, you can see why you can make compact cameras with fast lenses so small. The new Panasonic 20mm f/1.7 lens can get away with a mere 12mm of glass.
2. Aperture blades don't play a part when the lens is used wide open, but when you start closing down the aperture they come into play. A lens with only 5 blades will create more of a pentagon shape than a circle. Hence more expensive lenses will use more blades, and also make them curved. This has the effect of making out of focus areas more pleasing to the eye (bokeh). This is quite different than depth of field (the amount in focus) and is something people are willing to pay more for. A good example of this can be found at The Digital Picture where about half way into the review he compares the Canon 50mm lenses at the same aperture - with very different results in the out of focus areas.
3. If you purchase lenses you'll probably notice that cheaper ones have a variable aperture. A typical example is the kit lens you get with a DSLR - it will come with an 18-55mm zoom lens with an aperture rated from f/3.5 to f/5.6. At different focal lengths the aperture size makes for a different F number. As you zoom in the aperture hole becomes smaller relative to the focal length, and lets in less light. As well, not keeping a larger maximum aperture for the whole zoom range reduces the size of the lens, and ultimately the diameter of the lens and glass - reducing the price.
Let me know if this helps, or hurts your brain ;)