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.
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.
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 where and cancel each other out.
Formally, this means that both composites and are equal to identity morphisms (morphisms which 'do nothing' or declare an object equal to itself). That is, and .