What do I need to know about fitting new wheels on a bike?
There are a number of factors to consider when buying a new wheel set:


Wheel size, there are many different wheel sizes to choose from, modern road bikes are generally 700c and mountain bikes 26, 27.5 or 29inch. There's also a number of legacy, children's and specialty sizes.
Rim width, rim width determines how wide a tire you can run and is determined by seat stay and chain stay clearance. If you have a road bike you may be able to fit wider cross tires if you were so inclined.
Axle width, the width of the rear drop outs, generally 135mm on road bikes but older bikes may be less, 135 or 142 for mountain bikes. Front widths on mountain bikes are either 100 or 110.
Cassette size, you need a hub (unless using a free wheel) that is spaced for the cassette you'll be using. There are 7, 8/9/10 and 11 speed free hubs. A 7 speed cassette will fit an 8 speed hub with a spacer.
Cassette type, shimano and sram require a compatible hub, campy requires a different compatible hub.
Brake type, rim brakes need a specific rim, disc brakes and coaster brakes require a specific hub.
Axle diameter, on Mountain bikes there's 9mm QR and 15mm and 20mm thru axle on the front and 10mm QR, 10mm bolt up or 12mm thru axle on the rear.


Everything thing else such as material, number of spokes or valve type is user preference.