What is an Homomorphism/Isomorphism "Saying"?

One analogy I use in class says (loosely) that various kinds of morphisms (iso-, homo-) can be thought of as translations from one language to another.

An isomorphism provides a perfect translation in both directions. Words correspond one to one. Anything you can say in one language you can say equally well in the other.

A homomorphism can map many words in one language to the same word in another, effectively creating synonyms. There's an old saying that Eskimo languages have many different words for "snow". The Eskimo-to-English dictionary (homomorphism) would show that (it's not one-to-one, injective), and there would be words in English that didn't come from an Eskimo word (it's not onto, surjective).

Footnote: See THE GREAT ESKIMO VOCABULARY HOAX .

I'll consider groups. A homomorphism of groups (or an isomorphism) is simply a map between groups that preserves the structure of a group. What does this mean? Well, the structure we have on a group is an operation $*$. Then, to say that a map $\varphi$ preserves this structure is to say that multiplication in the domain translates to multiplication in the codomain. Formally, if

$$\varphi \colon G \to H$$

is a homomorphism of groups, then $\varphi (xy) = \varphi (x) * \varphi (y)$ for $x, y \in G$. If $\varphi$ is an isomorphism, then the groups (if they're finite) have identical multiplication tables (because $\varphi$ is a bijection and hence every product $h_1 h_2$ where $h_1, h_2 \in H$ corresponds to a product $g_1 g_2$ in our first group), which is what we would want if we say that groups are "equivalent".

One would perhaps want a homomorphism to preserve other structure between groups such as identity or inverses. In other words, we could wish for $varphi$ to map the identity in $G$ to the identity in $H$ (i.e. $\varphi (e_G) = \varphi (e_H)$) or we could want $\varphi$ to map inverses to inverses, meaning $\varphi (x^{-1}) = \varphi (x)^{-1}$. It happens that these properties are actually encapsulated in the definition of a homomorphism.