The purpose of wiring your tree is to shape it and give it a more appealing look and ultimately to imitate a full grown tree. Sometimes trees are shaped in an unnatural way just for the aesthetic aspect. Wiring can also be used for correcting the direction in which a branch or a leader* is growing.
For the wiring copper or aluminium wire is used, which comes in different thicknesses dependant of the size of the branch that needs to be wired. A good trick to decide what thickness of wire to use is to keep the wire against the branch and if you can move the branch with it, it should be thick enough to do the job.
*Leader: the leading branch that is the extension of the trunk