This is a question similar to: Understanding the Laplace operator conceptually
Homomorphisms and isomorphisms are easy to define:
1) given two groups $(G, *)$ and $(H, \cdot)$, a homomorphism is a function that maps elements from $G$ to $H$ such that if $a, b \in G \implies f(a*b) = f(a) \cdot f(b)$
2) a homomorphism $f$ is an isomorphism if $f$ is bijective
...but who cares? How would you explain why it is important to distinguish functions that fit these definitions? How do you wish you had been taught these concepts?
Any good essays (combining both history and conceptual understanding) on isomorphisms and homomorphisms that you would highly recommend?