An uncountable example is obtained by considering dense linear orders without endpoints. Any two dense linear orders without endpoints are elemenatarily equivalent. But $\langle\mathbb{R},<\rangle$ and $\langle\mathbb{I},<\rangle$ with the usual order, $\mathbb{I}$ the set of irrational numbers, are non-isomorphic as the first is complete while the latter is not.
There is no such countable example as the theory of dense linear orders without endpoints is $\aleph_0$-categorical.
I am interested in countable and finite examples of elementarily equivalent but non-isomorphic first-order structures.
Best Answer
You're not going to find any finite examples. For any finite structure $A$, every model of $\text{Th}(A)$ is isomorphic to $A$.
But there are many many countable examples. In fact, if $T$ is any complete theory which is not $\aleph_0$-categorical, then (by definition) it has countable models which are elementarily equivalent but not isomorphic.
Here are some concrete examples: