TITLE: What does it even mean to say 'preserve structure'?
QUESTION [9 upvotes]: Could somebody give a concrete example of a group structure being preserved in a isomorphism, et cetera? I always hear this 'preserve structure' thing. Ok, could somebody give me a rigorous definition of what it means to preserve structure? I'm tired of hearing things like 'they behave the same way under some operation bla bla bla'

REPLY [2 votes]: To complement Dorebell's excellent answer, I'll describe the same universal algebra ideas in the language of category theory rather than set theory. Well, I won't describe them, because other people have described them very well already, so I'll link existing expositions.
If you're not already familiar with the idea of a category, I recommend reading the opening pages of Awodey's Category Theory or Freyd and Scedrov's Categories, Allegories to become acquainted.
In any category that has finite products (and therefore, automatically, a terminal object), you can define a group object as an object $G$ equipped with a certain system of arrows between $G$ and products of $G$. Certain diagrams involving these arrows are required to commute. The arrows are the processes of multiplication, inversion, and producing the identity element. The commutative diagrams are the requirements that multiplication is associative, multiplying with the inverse gives the identity, and multiplying with the identity does nothing.
A very nice feature of this desciption is that it describes many different flavors of group in one breath. For example, a group object in the category of sets is an ordinary group, a group object in the category of smooth manifolds is a smooth Lie group, a group object in the category of supermanifolds is a supergroup, a group object in the category of algebraic varieties is an algebraic group, and a group algebra in the opposite category of commutative rings is a Hopf algebra (which doesn't even sound like it's a group). There are lots more examples where most of those game from.
Describing the "certain system of arrows" that defines a group object is pretty tedious, so people save time by calling this system of arrows a group structure. Then they can say, "Oh, group objects are simple! A group object is just an object equipped with a group structure."
Now we can finally say what it means for an arrow to "preserve structure." An arrow $f \colon X \to Y$ between two objects naturally induces arrows between their products: for example, we get an arrow $f \times f \colon X \times X \to Y \times Y$. An arrow $f \colon G \to H$ between two group objects preserves the group structure if $f$ and $f \times f$ and so on commute with the arrows giving the group structures of $G$ and $H$.
I've been talking about group objects as an example, but you can do the same thing with other kinds of algebraic structures. For example, you can study about ring objects and natural numbers objects. Natural numbers objects are super cool: just like group objects in different categories give different flavors of group theory, natural numbers objects in different categories give weird alternate-universe versions of number theory.