The BAND is a ring of metal with the bracket attached that is glued onto the tooth. This is what anchors the wires that will move the teeth.
The BRACKET or BRACE is the attachment cemented to the tooth or welded to the band. These hold the arch-wires.
The ARCH-WIRE is a removable, thin wire that fits into the bracket slots. This is what guides the teeth into place.
The ELASTIC LIGATURE is a plastic ring that ties arch-wire into the brackets.
The LIGATURE WIRE is a tiny wire that ties arch-wire into the brackets.
The HOOKS are used to attach elastics (rubber bands).