The brackets and arch wire work together to move the teeth into new positions that will correct your bite, straighten your teeth, correct gaps, etc. The arch wire puts pressure on the brackets with distribute the pressure to the entire tooth, and the orthodontist uses the arch wire to dictate how the teeth are moved into their new positions. This movement needs to be done slowly otherwise you risk losing your teeth. This is why braces are commonly worn for two to two and a half years and adjustments are only made every four to six weeks.
The reason this works is because the arch wires want to keep their original shape and will exert a great deal of force to return to their original arch shape. The materials used to make the arch wire increase their stiffness when exposed to your body heat. It is the combination of your body heat and the desire of the arch wire to keep its shape that provide the wire with enough strength to pressure your teeth to move.
Braces alone, however, are not always able to do the work by themselves. Sometimes you will have to wear also rubber bands to manipulate the pressure to pull the teeth in the correct direction.
Another part of the braces process is the Retainer, which needs to be worn after your braces are removed, to hold the teeth in their fixed positions.