In my lecture on mathematical logic, we said that the constructible universe $(\mathbb{L}, \in)$ is a model for the Axiom of Choice because all well-orderings are constructible, that is, definable by an FO-formula.
Later on in the lecture, it is shown, using the back-and-forth method to construct an isomorphism from a well-ordered structure to a non-well-ordered structure, that well-orderings are not FO-axiomatizable.
Intuitively, that seems like a contradiction to me. Could someone explain to me how well-orderings, or any class of sets, can be constructible but not axiomatizable?
The two notions are completely unrelated to one another.
Being constructible is a specific set theoretic property. It means that you are created by iterating pointwise "taking definable subsets", generally starting from the empty set. It has little to do with model theory, in general.
Being axiomatizable has little to do with set theory itself. It means that in a given first-order language which includes a binary relation symbol $<$, you are able to write axioms which guarantee that $<$ is interpreted as a well-ordering of the model.
In the case of well-orders, this is not a first-order property, and a quick ultrapower argument can show that any well-ordered structure is elementary equivalent to one which is not well-ordered. Therefore there is no theory which ensures the $<$ symbol is interpreted as a well-order.