The easiest way to explain aperture is like this... Think of your camera as an eye, and the aperture is the iris.
The darker the surroundings the more your eye opens up to let in more light, and that's exactly how the aperture works.
Aperture itself is counted by a system called the f-number, or f-stop. The slightly confusing part (or I thought it was) is that the numbers work backwards, the higher the number, the smaller the aperture.
A value of f/1 is the widest aperture and therefore the 'fastest' lens with the numbers dropping down to the 20s and 30s for the smallest opening.
Most, if not all, cameras allow you to change the aperture, any f-number that's on the lens of camera is purely it's maximum width, it can still go smaller.
Obviously the wider the opening, just like an eye, the more light let in, and vice versa.
Here's a little visual representation...
Both those pics were taken in the same conditions with the same shutter speed but the left hand one was with an aperture of f/5, the one on the right was at f/1.8.
You can see the vast difference in the two pictures thanks to all the extra light allowed in by the wide f/1.8 aperture.
You can also see on the box how most lenses (and this works for other cameras too) marks it's maximum aperture, in this case f/1.8... ignore the II after it, that's because this is the mark II version of the lens... just to add extra confusion 
The only thing to note when picking an aperture is sometimes it's not best to simply go for the widest option. Most lenses for DSLRs or compact cameras work best when shrunk by one or two 'stops'.
For example the lens I used for the above pics actually gives a sharper picture at f/2.5 than it does at f/1.8, so if the shutter speed allows it it's often best to let the eye close a little for a better photo...
Basically, your camera needs a bit of a squint.
Aperture also has a bearing in other things but I'll leave that for a different post 
rowtheboat
Interesting. And well explained.