If you think about it really deep, axioms are really just characteristics of the model we want to work with.
The model of the theory of rings is a field if it satisfies the axioms of a field, which require it not only to be a ring - but to be a commutative ring, without zero divisors and that every nonzero element is a unit.
The model of the theory of fields is algebraically closed if it satisfies the "closure" axiom, namely every polynomial formula has a solution in the field.
The model of ZF, $V$, is a model of ZFC if it also satisfies the axiom of choice, that every family of nonempty sets in the model has a choice function.
If you really think about these things, all those theorems starting with "Let $x$ be this object, with such and such properties. Then $x$ has property $\tau$." is really just to say that if $x$ is a model of some theory which we used, but is also a model of such and such additional axioms, then it is a model for the sentence $\tau$.
So to say that $X$ satisfies the second countability axiom, or the $T_3$ separation axiom is simply to say that we adjoin the theory two more axioms and that $X$ is a model of the larger theory and not just the original theory (topological space).
I once explained someone that if some property holds then we can prove the axiom of choice, he was quite baffled by the fact that we can "prove an axiom", but this is really all we do. We prove one property follows from another, or is equivalent to another, and we actually say "This additional axiom is consistent with the previous ones".
Added: What is a model? We start with a language, in the language we have all the things we want to have: functions, relations, constants, etc.
Now we can interpret this language in a structure. The structure is really just a nonempty set with the addition that we specify how to interpret every symbol of the language (this constant is this element; this predicate is this subset; etc.).
For example, we can interpret the language with a single binary relation as $\subseteq$ in the power set of $X$; or as $\le$ in the real numbers, those are obviously very different ways to understand the same language.
In the language we can write statements. Let us ignore for the moment from the first/second/higher order logic, and allow us to write pretty much anything. We can then write "Every collection of nonempty relations has a function which chooses elements such that these elements have such and such properties with the said relations", or we can write "There is some element that has a certain property".
Now consider the situation that we have a language, and we have an interpretation for this language, and this structure is such that some statement is true in it, then we say that it is a model for that statement.
So in short (after all this long introduction) a model for a theory is just a structure in which all the statements of the theory are true.