There are a real numbers worth of countable models of Peano arithmetic, all elementarily equivalent to the usual model, but all pairwise nonisomorphic.
Every nonstandard example (nonstandard means not isomorphic to the usual model) is characterized by the fact that it contains "infinitely large" numbers. That is, each model M has a canonical copy of N = usual naturals inside. However, each nonstandard model M has an element p (in fact, countably many such elements) which the model things is bigger than everything in its copy of N.
They are created by standard compactness arguments paired with the downward Lowenheim-Skolem theorem.
Let PA denote the axioms for Peano Arithmetic and set T = Th(N), the theory of N (i.e., the set of all first order statements which are true in the usual interpretation.
Finally, add a constant c to the language and let phi_n be the statement c > n, or more appropriately, c > SSSS.....S(0), where S is the successor function of PA and there are n S's in the expression.
Now, consider the set of axioms PA union T union phi_n for every n. By a standard compactness argument, this set of axioms is consistent, and so by the Godel completeness theorem, there is a model M of all the axioms simultaneously. By downward Lowenheim-Skolem, we can assume M is countable.
Now, ask yourself, what can c be? Well, since M satisfies phi_n for every n, we must have c > n for each "usual" natural number in M, i.e., c is infinite!
What's really cool is that by playing around with different kinds of phi_n, one can find models that have elements which are divisible by NO "standard" prime number. One can also find models that have elements which are simultaneously divisible by EVERY "standard" prime number!