Your bike's wheels play a major role in how it feels to ride. After the frame material and construction, the design, purpose and make-up of your wheels is probably the biggest factor in deciding the speed, efficiency and feel of your bike. Because of this, you might consider a wheel upgrade even if your existing wheels aren't worn out, broken or buckled.
Road wheels are usually focused around speed. Many riders quickly upgrade from the wheels their bike came with to fit lighter, stiffer, more aerodynamic wheels.
Because most road bikes have rim brakes, the braking surface will gradually wear, so eventually the wheel will need replacing before the wall of the rim becomes dangerously thin. Top-level race wheels can be built from Carbon as well as Aluminium. Deep section wheels are designed for an aero advantage.
At the other end of the scale, you may prefer to choose a pair of sturdier wheels for a bike used for commuting or training. Tougher wheels will have a higher spoke count - up to 36 - to make them more durable. You can now even get 700c wheels with a disc brake hub for disc-braked road bikes or cyclocross bikes.
Although road bikes are generally compatible in axle width and type, you need to make sure your rear wheel has a compatible freehub. Your bike's gear cassette fits onto the freehub body, so you need to make sure it is compatible. The wheel description should list the cassette type (e.g. 10-speed) and sometimes the brand as well (i.e. Shimano, SRAM or Campagnolo).
Finally, consider what kind of tyres you will be fitting. Whether you go for clincher, tubular or tubeless tyres, you'll need to make sure you have the right sort of rims on your new wheels to accommodate them.
If you ride off road, your wheels could be prone to damage from rock strikes or crashes. You might also want to upgrade to improve performance. MTB wheels are available in three sizes - 26in, 27.5in (650b) and 29in. Make sure you choose the right size. Most mountain bike wheels now have disc brake hubs, but you still need to make sure your new hubs are compatible with the type of disc brakes you will be using.
Disc rotors will be fitted to the hubs with six-bolt fittings or Shimano CenterLock. If your new wheels aren't compatible with your rotors, you can buy adaptors to convert them. There are quite a lot of different axle standards for mountain bikes, so you need to make sure your new wheels and axles will fit your bike's frame. If in doubt, it's best to stick to like-for-like, or contact firstname.lastname@example.org for help choosing.
26in wheels used to be the standard on mountain bikes, but have now been somewhat superseded by 29er and 650b wheels. Many riders still prefer the level of acceleration and control you get with them though, and you'll find them on older mtbs and bikes with smaller frames such as women's mtbs.
27.5in (650b) wheels offer a great balance of fast rolling and control on difficult terrain. They are fast becoming the most popular wheel size being preferred over the other 2 sizes of 26 & 29 inch options.
29in wheels feel stable and roll fast over rough terrain. They are particularly favoured by taller riders.
When replacing wheels on a hybrid, you can just choose road or mountain bike wheels in the right size, with the right size and hub compatibility. There are some hybrid specific wheels to choose from. These tend to be strong, budget friendly wheels built to withstand everyday life.
When you choose your new wheels, don't forget the extras you might need to go with them. If everything's compatible, you can of course re-use your existing QR or bolt-through skewers, but you might want to upgrade at the same time. Also, not all wheels come with rim tape to line the wheels with, so you might need to buy this as well.