TITLE: How is the notion of adjunction of two functors usefull?
QUESTION [7 upvotes]: Is there a secret or an intuitive idea behind the fact of creating the concept of adjunction of two functors ( Functor - Adjoint Functor ) ? How is this notion of adjunction of two functors usefull ?
Thank you in advance for your help.

REPLY [10 votes]: I have at least two answers for this.
Every adjunction prepares an equivalence of categories.
Namely, if $(F,G,\eta,\varepsilon) : \mathcal{C} \to \mathcal{D}$ is an adjunction, let $\mathsf{Fix}(\eta) \subseteq \mathcal{C}$ denote the full subcategory of objects $X \in \mathcal{C}$ such that $\eta(X) : X \to G(F(X))$ is an isomorphism. We define $\mathsf{Fix}(\varepsilon) \subseteq \mathcal{D}$ analogously. Then, $F,G$ restrict to an equivalence of categories
$$\mathsf{Fix}(\eta) \simeq \mathsf{Fix}(\varepsilon).$$
This is also useful in the case of preorders, where adjunctions are known as Galois connections. Many famous equivalences of categories arise this way (and this seems to be kept as a secret).

The main theorem of Galois theory
Hilbert's Nullstellensatz
The theorem of Gelfand-Naimark
The duality between affine schemes and commutative rings
The duality of finite-dimensional vector spaces
Pontrjagin duality
Grothendieck's main theorem Galois theory
...

Of course, in all these theorems, the hard part is to determine the fixed objects. But the framework for these theorems is provided by basic adjunctions.
Left adjoints are cocontinuous.
That is, if $F$ is left adjoint to a functor $G$, then $F$ preserves colimits. This is a standard fact, and it is easy to prove, but it subsumes hundreds of concrete isomorphisms which you have (or will) come across. For example, $$S^{-1}(\bigoplus_i M_i) \cong \bigoplus_i S^{-1} M_i$$ (localization of modules) is a formal consequence, as is $$(G * H)^{\mathrm{ab}} \cong G^{\mathrm{ab}} \oplus H^{\mathrm{ab}}$$ (abelianization of groups). We don't have to compute anything to get these isomorphisms.
The tensor product is right exact since tensoring with some module has a right adjoint, the (internal) hom functor. We don't have to fiddle around with cokernels, which is done in many texts which avoid adjoints and prove right exactness directly.
There is even a sort-of converse of this general result, namely Freyd's general adjoint functor theorem: Every cocontinuous functor, which has a solution set (which is a quite mild set-theoretic condition), is a left adjoint. So the preservation of colimits by left adjoints (and, dually, of limits by right adjoints) is perhaps one of their main goals.