Examples of adjunctions
Free constructions: given a set you get a free groupmonoidring/vector space/etc. - each of these is a left adjoint. The corresponding right adjoint throws away the algebraic structure.
Given a graph you get a free preorder or a free category, the corresponding right adjoint is the underlying graph of a preorder/category.
Discrete things: given any set you get a discrete preordergraphmetric spacecategorytopological space/etc.; each of these is a left adjoint and the corresponding right adjoint again recovers the set.
Codiscrete things: given any set you get a codiscrete preorder, complete graph, codiscrete category, category, etc.; each of these is a right adjoint and the left adjoint gives the underlying set.
Given a group, you can quotient by its commutator subgroup to get an abelian group; this is a left adjoint. The right adjoint is the inclusion of abelian groups into Grp