Every field of mathematics has a different meaning of dual. Loosely, where there is some binary symmetry of a theory, the image of what you look at normally under this symmetry is referred to as the dual of your normal things.
In linear algebra for example, for any vector space V, over a field, F, the vector space of linear maps from V to F is known as the dual of V. It can be shown that if V is finite-dimensional, V and its dual are isomorphic (though no isomorphism between them is any more natural than any other).
There is a natural embedding of any vector space in the dual of its dual:
V -> V'': v -> (V': w -> wv : F)
(x' is normally written as x with a horizontal bar above it). I.e. v'' is the linear map, from V' to F, which maps any w to the scalar obtained by applying w to v. In short, this double-dual mapping simply exchanges the roles of function and argument.
It is conventional, when talking about vectors in V, to refer to the members of V' as covectors.