As you know, the Commonwealth government can only make laws that it has been given the power to make under s 51 of the constitution.
Sometimes an individual, business or State will challenge a Commonwealth law, claiming that the Federal Parliament did not have the power to pass it, since it is not a law "with respect to" a given subject matter under s 51. So the High Court must interpret both the impugned law and the section of the constitution in question to resolve the matter.
'Characterisation' in Australian constitutional law is thus the process of determining whether a law comes within a head of power under s 51.