A pair of mathematical structures are isomorphic to each other if they are "essentially the same", even if they aren't necessarily equal.

An isomorphism is a morphism between isomorphic structures which translates one to the other in a way that preserves all the relevant structure. An important property of an isomorphism is that it can be 'undone' by its inverse isomorphism.

An isomorphism from an object to itself is called an automorphism. They can be thought of as symmetries: different ways in which an object can be mapped onto itself without changing it.

Equality and Identity

The simplest isomorphism is equality: if two things are equal then they are actually the same thing (and so not actually two things at all). Anything is obviously indistinguishable from itself under whatever measure you might use (it has any property in common with itself) and so regardless of the theory or language, anything is isomorphic to itself. This is represented by the identity (iso)morphism.

Group Isomorphisms

For a more technical example, the theory of groups only talks about the way that elements are combined via group operation. The theory does not care in what order elements are put, or what they are labelled or even what they are. Hence, if you are using the language and theory of groups, you want to say two groups are essentially indistinguishable if you can pair up the elements such that their group operations act the same way.

Isomorphisms in Category Theory

In category theory, an isomorphism is a morphism which has a two-sided inverse function. That is to say, is an isomorphism if there is a morphism $g : B \to A$ where $f$ and $g$ cancel each other out.

Formally, this means that both composites $f g$ and $g f$ are equal to identity morphisms (morphisms which 'do nothing' or declare an object equal to itself). That is, $g f = {i d}_{A}$ and $f g = {i d}_{B}$ .