Why should I care about adjoint functors I am comfortable with the definition of adjoint functors. I have done a few exercises proving that certain pairs of functors are adjoint (tensor and hom, sheafification and forgetful, direct image and inverse image of sheaves, spec and global sections ect) but I am missing the bigger picture.
Why should I care if a functor has a left adjoint? What does it tell me about the functor? 

For one thing, it tells you that the functor respects colimits. 
For example, the "free group" functor is the left adjoint of the "underlying set" functor from $\mathcal{G}roup$ to $\mathcal{S}et$. The fact that it is a left adjoint tells you that it respects colimits, so the free group of a coproduct is the coproduct of the free groups. The "coproduct" in $\mathcal{S}et$ is the disjoint union, and the coproduct in $\mathcal{G}roup$ is the free product: so the free group on a disjoint union $X\cup Y$, $F(X\cup Y)$, is (isomorphic to) the free product of the free groups on $X$ and $Y$, $F(X)*F(Y)$. 
Dually, right adjoints respect limits; so in the case above, the underlying set of a product of groups is the product of the underlying sets of the groups.
Added: Ever wondered why the underlying set of a product of topological spaces is the product of the underlying sets, and the underlying set of a coproduct of topological spaces is also the coproduct/disjoint union of the underlying sets of the topological spaces? Why the constructions in topological spaces always seem to start by doing the corresponding thing to underlying sets, but in other categories like $\mathcal{G}roup$, $R-\mathcal{M}od$, only some of the constructions do that? (I know I did) It's because while in $\mathcal{G}roup$ the underlying set functor has a left adjoint but not a right adjoint, in $\mathcal{T}op$, the underlying set functor has both a left and a right adjoint (given by endowing the set with the discrete and indiscrete topologies).
