You need wheels for your bike, otherwise you won't go very far with out them. For a long time they were made of steel and were heavy. Wheels had 40 or more spokes and were hand made and would take time for a trained wheel builder to put together.

Alloy came along and hubs and rims got much lighter, also as the rims got stronger they didn't need so many spokes, so they went from 40 to now down to 16, thus saving more weight and lessening wind drag.
Spoke patterns have changed a lot too, they used to cross four time for a soft, comfortable ride, now they only cross once or are radial and they are spaced in groups of three, like the new Campagnolo G3 wheels.
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Rims are either tubular or clincher type tyres. The tubular is stuck on to the rim with very strong glue and is a more responsive and lighter choice.
The clincher has a tyre and a separate inner tube, the rim is heavier but if you are not racing its not a problem and many teams race on clinchers now anyway. Rims can be shallow, medium or deep section and made of alloy or carbon and are for different road conditions and dependent on wind and weather.
Hubs are now mostly made of alloy but they are now some of carbon construction and have quick releases so you can change the wheel in a hurry.
For time trials or on the track you can have disc wheels or tri-spoke, all of these are carbon bicycle wheels and are for aerodynamics not for comfort.