Often in mathematics one has a set of objects, such as positive integers, matrices, groups, or surfaces, and a method of transforming one of these objects into another. And sometimes one has two objects and would like to prove that the first cannot be transformed into the second. One of the best ways of doing this is to find an invariant: that is, a quantity that does not change when you transform in the given way, but which is different for the two objects.
Links to articles about invariants
If the notion of an invariant is new to you, then you may wish to start with the following introductory article.
See also the Wikipedia article on invariants.
The articles below concern invariants in more specific contexts.