Intuition behind the definition of Adjoint functors I think of adjoint functors as some sort of inverses. So, the first part of the definition looks reasonable that there exists natural transformations 
$$\epsilon : FG \rightarrow 1_C$$
$$\eta : 1_D \rightarrow GF$$
But what is the motivation behind the second part of definition?
$$1_F = \epsilon F \circ  F \eta $$
$$1_G = G \epsilon \circ \eta G$$
What does the second part of definition mean intuitively?
 A: These two triangle identities say that $\epsilon$ and $\eta$ are "inverse" to each other. More precisely, the Yoneda Lemma implies that $\epsilon$ corresponds to a natural transformation $\hom(x,Gy) \to \hom(Fx,y)$ and that $\eta$ corresponds to a natural transformation $\hom(Fx,y) \to \hom(x,Gy)$. Unwinding these correspondences one checks directly that these transformations are inverse to each other iff $\epsilon$ and $\eta$ satisfy the triangle identities.
There are several ways to think about adjunctions. See the links in the comments. There are quite geometric interpretations, for example via string diagrams, see the nlab. Also, the classifying space of a category sets up a connection between category theory and topology (and basically this is the reason why higher category theory has fused with topology):


*

*Category <-> Space

*Object <-> Point

*Morphism <-> Path

*Endomorphism <-> Loop

*Functor <-> Continuous map

*Natural Transformation <-> Homotopy

*Adjunction <-> Homotopy equivalence

*initial/terminal object <-> Contraction


It is also important to recoqnize that adjunctions often "prepare" equivalences. More precisely, if $F : D \to C $ is left adjoint to $G : C \to D$, then one can define the fixed points of the endofunctor $FG$ as those objects of $C$ where $\epsilon$ is an isomorphism, and analogously the fixed points of $GF$ as those objects of $D$ where $\eta$ is an isomorphism. The triangle identities imply that $F$ and $G$ induce an equivalence of categories between the fixed points of $FG$ and the fixed points of $GF$. Lots of equivalences of categories can be seen that way, for example the Main Theorem of Galois theory, Pontrjagin duality, Gelfand duality, Stone duality, the duality between commutative rings and affine schemes, the classification of coherent sheaves on projective spaces, the (weak) equivalence between CW complexes and Kan simplicial sets, and so on.
