What exactly does it mean to say a map is "structure preserving"? And what is the difference between this and a isomorphism? 

f:G -> H. 

f(g1 ∗ g2) = f(g1) ∗′ f(g2) . is the definition of a homomorphism. Which seems to say that you can transfer the binary algebraic structure in G to a binary algebraic structure in H. But intuitively, what does that really mean? I have no taken that many math classes, so if you could keep the examples you may use to as basic and primitive as possible, that would be terrific!