Braces work by exerting gentle pressure on the teeth which, over time, makes them move to a new position. They can be divided into two main types, removable and fixed. A removable brace, as the name implies, can be taken out for thorough cleaning and for occasions when you do not wish to wear it. A fixed brace on the other hand, is fixed to your teeth and cannot be removed for the duration of your treatment.
There are several parts to a fixed brace which helps your teeth move to their new position, the bracket, which is either bonded to the tooth or to a band around a tooth, an arch wire and something to exert a force on the bracket such as an elastic.
Taking each element separately, brackets will either be glued to the front or back of your teeth (see the post on lingual braces for more details about having them on the back of your teeth) or will be attached to a metal band which fits around your teeth (usually the back teeth). The purpose of the bracket is to provide something firmly fixed to each tooth so that it can be moved to a new position.
An arch wire is a thin piece of flexible wire which runs through a slot on each bracket. The wire is held onto the bracket either by small elastic O-rings such as on the photo or by a mechanism on the bracket to prevent the wire from becoming detached. The arch wire will exert a gentle pressure on each tooth to move it to a new position. As your treatment progresses, different arch wires will be used to guide your teeth into their new position so you will find your orthodontist replacing the wire on some of your appointments.
The final component is something to exert an additional force on your teeth such as elastics. These can help pull teeth together along an arch wire, move individual teeth or guide your jaw into a new position.
This is an interesting clip on how braces work.