Do the axioms of set theory actually define the notion of a set? In Henning Makholm's answer to the question, When does the set enter set theory?, he states:


  In axiomatic set theory, the axioms themselves are the definition of the notion of a set: A set is whatever behaves like the axioms say sets behave.


This assertion clashes with my (admittedly limited) understanding of how first-order logic, model theory, and axiomatic set theories work.
From what I understand, the axioms of a set theory are properties we would like the objects we call "sets" to have, and then each possible model of the theory is a different definition of the notion of a set. But the axioms themselves do not constitute a definition of set, unless we can show that any model of the axioms is isomorphic (in some meaningful way) to a given model.

Am I misunderstanding something? Is the definition of a set specified by the axioms, or by a model of the axioms? I would appreciate any clarification/direction on this.



Update: In addition to all the answers below, I have written up my own answer (marked as community wiki) gathering the excerpts from other answers (to this question as well as some others) which I feel are most pertinent to the question I originally posed.
Since it's currently buried at the bottom (and accepting it won't change its position), I'm linking to it here. Cheers!
This is the commonplace clash between the semi-Platonic view of the laymathematician and the foundational approach for mathematics through set theory.

It is often convenient, when working in "concrete" mathematics, to assume that there is a single, fixed universe of mathematics. And everyone who took a course or two in logic and set theory should be able to tell you that we can assume this universe is in fact a universe of $\sf ZFC$.

Then we do everything there, and we take the notion of "set" as somewhat primitive. Sets are not defined, they are just the objects of the universe.

But the word "set" is just a word in English. We use it to name this fickle, abstract, primitive object. But how can you ensure that my intuitive understanding of "set" is the same as yours?

This is where "axioms as definitions" come into play. Axioms define the basic ground rules for what it means to be a set. For example, if you don't have a power set, you're not a set: because every set has a power set. The axioms of set theory define what are the basic properties of sets. And once we agree on a list of axioms, we really agree on a list of definitions for what it means to be a set. And even if we grasp sets differently, we can still agree on some common properties and do some math.

You can see this in set theorists who disagree about philosophical outlooks, and whether or not some conjecture should be "true" or "false", or if the question is meaningless due to independence. Is the HOD Conjecture "true", "false" or is it simply "provable" or "independent"? That's a very different take on what are sets, and different set theorists will take different sides of the issue. But all of these set theorists have agreed, at least, that $\sf ZFC$ is a good way to define the basic properties of sets.



As we've thrown Plato's name into the mix, let's talk a bit about "essence". How do you define a chair? or a desk? You can't really define a chair, because either you've defined it along the lines of "something you can sit on", in which case there will be things you can sit on which are certainly not chairs (the ground, for example, or a tree); or you've run into circularity "a chair is a chair is a chair"; or you're just being meaninglessly annoying "dude... like... everything is a chair... woah..."

But all three options are not good ways to define a chair. And yet, you're not baffled by this on a daily basis. This is because there is a difference between the "essence" of being a chair, and physical chairs.

Mathematical objects shouldn't be so lucky. Mathematical objects are abstract to begin with. We cannot perceive them in a tangible way, like we perceive chairs. So we are only left with some ideal definition. And this definition should capture the basic properties of sets. And that is exactly what the axioms of $\sf ZFC$, and any other set theory, are trying to do.
Every axiomatic system begins with undefined concepts, definitions, and axioms. For example, in geometry, some of the undefined concepts are "point", "line", and "incidence".  

Likewise, in set theory, some undefined concepts might be "element", "set", and "is a member of".  The peculiar thing about undefined concepts is that when you do a dictionary search, you always get back to the thing you are looking up if you look into synonyms of synonyms of synonyms.   There is no way to get a hard definition.   Nevertheless, we all know what element, set, and membership mean.