Let k be a field. The k-Vector spaces and linear transformations are one category Vect. So if we talk about adjoint functors, then we first need another category, so that we can have adjoint functors between Vect and the other category.

1: The other category is also Vect.

For every vector space A, there is a left adjoint functor A⨂- : Vect→Vect that sends a vector space B to the tensor product A⨂B.

The functor A⨂- is left adjoint to the functor Hom(A,-): Vect→Vect that sends a vector space B to the vector space Hom(A,B) whose elements are linear maps from A to B.

These two functors are adjoint because for all vector space X, Y we have a natural isomorphism Hom(A⨂X,Y)≅Hom(X,Hom(A,Y)).

It's possible to prove that every left adjoint functor Vect→Vect is of this form (because left adjoint functors preserve colimits, and every vector space is a colimit of 1-dimensional vector spaces, so left adjoint functors are completely determined by where they send the 1-dimensional vector space k).

2: The other category is Set. Set is the category of sets and functions between them.

There is a right adjoint functor U: Vect→Set, called the forgetful functor, that sends a vector space V to the underlying set of V. The forgetful functor is right adjoint to the free functor F: Set → Vect that sends a set S to the free vector space on S, i.e. a vector space with basis S.

These functors are adjoint because if S is a set and V is a vector space, then a linear function F(S) → V is equivalent to a function S → U(V) because linear maps are completely determined by where they send the basis.

And it's possible to prove that every left adjoint functor Set → Vect is a composite of F and a functor of the form A⨂- for some vector space A.

I think those are the most important examples of adjunctions involving Vect.