I know that a mapping $\phi:A\to B$ is a homomorphism provided that $$\phi(A*B)=\phi(A)\times\phi(B)$$ where $*$ and $\times$ are two operators on the algebraic structures $A$ and $B$ respectively. In other words, these mappings preserves structure of the operations.
I am, however, trying to get a deeper understanding of homomorphism. If they preserve the structure, what does this mean? What can we do with this?
An extention of homomorphisms, are isomorphism in which the two structures can then be seen "as the same". I can see how this can be a powerful tool, but without explicitly knowing the one-to-one and onto properties, what can homomorphisms be used for in general?