An ordinary nut with an ordinary washer, used to fasten a bolt on something that is subject to repetitive vibration--say, a wheel axle or engine accessory--would eventually vibrate loose. Lock washers and locknuts prevent this from happening. Lock washers, when tightened to the required torque beneath an ordinary fastener, will exert a spring tension against the fastener to keep it from vibrating loose. Locknuts typically have a malleable insert that deforms to grip the threads of the bolts they're threaded on to, which in turn keep the nuts from vibrating loose. Most projects will call for either locking washers or locking nuts--not both--to be used on a given fastener.
Place the lock washer between the nut and the work surface if using a nut-and-bolt configuration, or between the fastener head and the work surface if using a tapped hole configuration (in other words, the bolt goes straight into the hole and doesn't poke out the other end, so no nut is used).
Verify that the teeth of the locking washer engage completely with both the fastener (or nut) head and the work surface. If the teeth on the washer don't engage with both surfaces, it won't hold. Change sizes or styles of lock washer as needed to ensure the washer's teeth mate with the work surface and fastener head before proceeding. If you're using a split washer, skip this step.
Tighten the nut (nut-and-bolt assemblies) or bolt (tapped hole assemblies) to the torque specified in your project manual. Use a torque wrench to ensure that the appropriate torque is achieved.
Inspect the locking washer to, once again, verify that the teeth are completely hidden by the fastener head and the work surface. If you're using a split (also known as helical) washer, inspect the washer to confirm that it is still "split," with one side out of alignment with the other so that the washer exerts spring tension on the fastener. If the required torque flattens the split washer into the shape of a standard washer, it will no longer function as a locking washer; use a toothed locking washer instead.
Place the locknut against the threaded end of the bolt, just as you would place an ordinary nut. Make sure that if the nut has a raised centre portion this faces out; the flat surface of the locking nut should sit flush against the work surface once tightened.
Hand-tighten the locknut until it's securely threaded on to the bolt.
Tighten the locknut with a torque wrench until it reaches the specified torque.
The most common sort of locking nuts have a nylon insert that deforms to grip the threads of the fastener, so they're meant to be used once, then discarded and replaced with a new locking nut; using this sort of nut multiple times may decrease its holding power as the already-deformed nylon will not fully grip the threads on repeat uses.