Wheels are what you need to get moving on you bike. There are two main types of wheels - traditional spoked wheels and fancy schmancy carbon fibre one piece exotic thingos. I prefer traditional wheels, which are really a combination of 6 components - starting from the center - the hubs, the spokes, the nipples, the rims, the rims strips(to protect the tube from the nipples), the tubes, and the tires (tyres for UK).
Spoked wheels are very light and strong when properly built, and are IMHO one the most important inventions of the past 500 years, right up there with the bicycle and sliced bread. They are produced by a variety of companies including Sun, Mavic, Spinergy, Rolf, and Shimano.
The basic theory behind traditional wheels is that you take a round thing called a rim, wrap a tire around it, and then make it spin with a thing caled a hub. The whole thing is held together by spokes, which are secured at the hub end by knobs on the end of the spokes and at the rim end by the nipples, which attach to the spokes with threads and to the rims with knobs. To keep the wheel straight, there are two sets of spokes, (usually) 16 pulling to the left and 16 pulling to the right. This creates a 32 spoke wheel. These spokes often overlap their neighbours to increase stiffness. The tension of the spokes is adjusted by using a spoke wrench, AKA a nipple twister.
To learn more about how to build wheels, you'll soon be able to visit my node on wheel building, coming soon to a theatre near you. For more information on fixing wheels and bikes in general, stay tuned, I'm working on it through my Bike Mechanic node.