Your braces are made up of several parts.

The bracket is an orthodontic attachment secured to a tooth for the purpose of attaching an arch wire.
The archwire is a metal wire which is attached to your brackets to move your teeth.
An elastic is a small rubber band that is attached at two points between the top and bottom teeth.
A band is a thin metal ring, usually stainless steel, which serves to secure orthodontic attachments to a tooth.
The coil spring fits between brackets and over the arch wire to widen spaces between teeth.
The elastic tie or rubber band is the tiny rubber band that fits around the bracket and holds the archwire in place.
The headgear tube is a round, hollow attachment where the inner bow of the headgear is inserted.
A hook is either welded or temporarily attached to the bracket.
A loop is created in your archwire to typically close spacing between teeth that has been cause by an extraction.
A tie wire is a thin piece of wire that is wrapped around the bracket to hold the archwire in place.