What is an isomorphism? I'm familiar with the concepts of group isomorphism, ring isomorphism, and graph isomorphism, but it's never been presented to me what an isomorphism is in general: given any X, what is an X isomorphism?
Informally, I understand isomorphism as "preservation of structure", where "preservation" is domain specific. Is there a formal definition?
 A: Suppose you have two structures $A$ and $B$, and suppose you call a function $f:A\to B$ "good" if it satisfies certain properties (for $A,B$ topological spaces, we might want $f$ continuous, for $A,B$ vector spaces we might $f$ to be a linear transformation, &c). Suppose that $f$ is invertible, and $f^{-1}$ is also "good". Then we say $f$ is an isomorphism. In the setting say of vector spaces, rings or groups, whenever $f$ is invertible, it is automatically "good" too, so sometimes people define an isomorphism to be an "invertible homomorphism" say when talking about groups or rings. However, one can give counterexamples when $A,B$ are topological spaces: there exists invertible continuous $f:A\to B$ such that $f^{-1}:B\to A$ is discontinuous.  
A: As the comment suggests, looking up the word "morphism" in the context of "Categories" and Objects will give you a more rigorous and general idea about isomorphisms. But I always like to think of isomorphism as something that allows you to make copies of a given "object", be it rings or groups, or Topological spaces or manifolds,etc. 
Conveniently you can also look at it as a transformation that allows you to preserve the algebraic / geometric / topological structure or atleast the most basic and desired properties pertaining to these.
Edit: The word "copies" might be very misleading actually. Its more prevalent in the context of "Product spaces" than here. But I could not find a better word.Actually I think it might be quite wrong to say copies. Instead I would go along with saying that it allows you to collect spaces which though not essentially copies of the original space still basically carry the same "bloodline" if you will. I realise that this is not a technically sound answer, but was trying to see how best I could put this without resorting too many technical terms.I apologise if this isn't what you need
