A few (mostly trivial) examples (may be related to references in Tim Porter's answer?):

First of all, the self-duality of the category of vector spaces can be enriched to other self-dualities:

(Of course, now we can generalize this to Hopf algebras.)

Another simple class of examples comes from taking the dual of some object, and asking what additional structure is necessary to reconstruct the original object. For instance:

- Let $V$ be a vector space. Consider $V^*$. It has a natural topology, in which open subspaces are orthogonal complements of finite-dimensional subspaces of $V$. (This is
essentially weak topology if $V$ is viewed as a discrete space.) This gives a duality
between vector spaces and certain class of topological vector spaces that can be easily described explicitly. (A fancier way to view this: vector spaces are 'ind-finite-dimensional', so their dual are 'pro-finite-dimensional'.)

This can be modified to duality on Tate vector spaces, if you happen to like this sort of things.

There are also some examples in (algebraic) geometry (that are probably not as straightforward as the others).

Finally, there is a (somewhat cheap) trick to start with a duality on the derived category, then take the abelian category sitting inside it, and realize that it is dual to its image. For instance:

Look at the derived category of constructible sheaves (say, on a scheme); it has Verdier's duality. Then the abelian subcategories corresponding to dual perversities are dual to each other. For instance, the category of constructible sheaves is dual to perverse sheaves for a certain perversity.

Look at the derived category of coherent sheaves on a scheme; it has Serre's duality. Then the abelian subcategories corresponding to dual perversities are dual to each other. For instance, the category of coherent sheaves is dual to perverse sheaves for a certain perversity.