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? 

According to MacLane, the slogan is, "Adjoint functors arise everywhere." So there is a good reason to study them!
Having an adjoint tells you that the functor commutes with (either) limits or colimits. If a functor has a left adjoint, then it commutes with colimits, while if it has a right adjoint, it commutes with limits. For nice categories, one can sometimes conclude the converse.
One example of this is in an abelian category. In the case of $R$-modules, for instance, the adjunction between Hom and the tensor product shows that the tensor product is right-exact (a.k.a. commutes with finite colimits). This is a somewhat more conceptual argument than the usual one.
Another example is the following. Let $X$ be a locally compact Hausdorff space. Then the functor $Z \mapsto Z \times X$ has an adjoint (namely, the functor $Y \mapsto Y^X$). It follows that taking products with $X$ preserves push-out diagrams, and more generally all colimits. This is useful sometimes in algebraic topology. For instance, if you have a push-out $A \cup_B C$ and homotopies $A \times I \to Z$ and $C \times I \to Z$ that agree on $B \times I$, you get a homotopy $(A \cup_B C) \times I \to Z$, the continuity of which might not be immediately obvious otherwise.
