I think of Gabriel-Ulmer duality as one of the early key results in categorical model theory.
Many structures in mathematics are what are called "essentially algebraic". This includes all algebraic structures (which model theories given by a functional signature and universally quantified equations): groups, rings, Lie algebras, etc. It also includes algebra-like structures which don't quite fit in this mold, for example categories, where some of the operations may only be partially defined (but the domains of the operations are given by equations involving other operations). There are various nice ways of expressing the syntax of an essentially algebraic theory: one is via limit sketches, another is just to use finitely complete categories, in much the same way that (following Lawvere) classical algebraic theories can be expressed as certain categories with products.
If we think of a small finitely complete category as a "theory" $T$, then a model of $T$ in this way of thinking is a finitely continuous functor $T \to Set$. The category of models would then be the category $Cont(T, Set)$ of such functors and transformations between them.
What is amazing and very nice is that the theory $T$ can be recovered from the category of models $M$, by considering functors
$$M \to Set$$
which preserve all limits (have left adjoints) and all filtered colimits. This is essentially Gabriel-Ulmer duality, giving a perfect dual correspondence between theories or syntax of finitary essentially algebraic type (small finitely complete categories) and semantic categories of models of such (which can be described in pure category terms as locally finitely accessible categories).
This perfect duality between syntax and semantics is part of a long story in categorical model theory, culminating the theory of accessible and locally presentable categories. Spiritually, it is reminiscent of many other dualities such as the Tannaka-Krein duality mentioned by Finn Lawler earlier. (I'll add some links to this later.)