How to recognize adjointness?
I think one can give not just one but several intuitive descriptions of adjunctions, some of which are more appropriate for understanding some adjunctions than others. For this reason it is perhaps less useful to ask someone else to describe their intuitions than to just collect a list of examples and build your own intuition from that list. But let me try anyway.
First, here is an overarching meta-intuition:
An adjoint is the next best thing to an inverse.
If a functor has an inverse, that inverse is necessarily both its left and its right adjoint (exercise). But being invertible is a very strong condition on a functor, and most functors are not invertible. However, many functors have adjoints, which roughly speaking means it's possible to construct a "best attempt at an inverse" (of which there are two, the left adjoint, or "best attempt from the perspective of maps out," and the right adjoint, or "best attempt from the perspective of maps in.")
Second, here is the simplest example I know where one can see the above intuition at work:
The inclusion $\mathbb{Z} \to \mathbb{R}$ of posets is not invertible, but it has both a left and a right adjoint given by taking the ceiling resp. the floor of a real number (exercise).
Next, here is a combination of list of other somewhat more specialized intuitions and other examples fitting those intuitions. Some of these overlap, and I make no claim that this list is exhaustive even of the examples that I know.
Forgetful / free adjunctions. In this type of adjunction, one of the functors is "forgetting" some kind of structure and the other constructs the "free" structure. Archetypal examples include the forgetful functor $\text{Grp} \to \text{Set}$ (whose left adjoint constructs the free group on a set), the forgetful functor $\text{Ring} \to \text{Grp}$ given by taking the multiplicative group (whose left adjoint construct the group ring on a group), or the forgetful functor $\text{Alg} \to \text{Lie}$ given by taking the commutator bracket (whose left adjoint constructs the universal enveloping algebra on a Lie algebra). But there are many others. A categorical property that separates this case from others is that forgetful functors are usually faithful.
Restriction (pullback) / extension (pushforward) adjunctions. In this type of adjunction, you're dealing with categories of objects "living on" other objects $X$, such that maps $f : X \to Y$ (and in particular inclusions) induce restriction functors from objects living on $Y$ to objects living on $X$. Often these restriction functors have both a left and a right adjoint, which are different kinds of "extensions" of an object living on $X$ to an object living on $Y$. These adjunctions come in both algebraic and geometric flavors.
Algebraic example: restriction functors $S\text{-Mod} \to R\text{-Mod}$ on categories of modules coming from morphisms of rings $f : R \to S$ have both left and right adjoints. The left adjoint is called extension of scalars or induction (depending on whether $f$ is more like an extension of fields or like an inclusion of group rings), and the right adjoint can be called coextension or coinduction, I guess.
Geometric example: restriction functors $\text{Sh}(Y) \to \text{Sh}(X)$ on categories of sheaves coming from morphisms of spaces $f : X \to Y$ (edit: sometimes) have both left and right adjoints. The left adjoint is called shriek pushforward and the right adjoint is called star pushforward.
See also Kan extension.
Hom / tensor adjunctions. These come in roughly two flavors, a more algebraic and a less algebraic flavor. The more algebraic flavor is exemplified by the adjunction between $(-) \otimes M$ and $\text{Hom}(M, -)$ where $M$ is a bimodule and I've suppressed mention of all of the underlying rings. The less algebraic flavor is exemplified by the adjunction between $(-) \times X$ and $\text{Hom}(X, -)$ where $X$ is a set or a space. The underlying idea is currying.
See also Cartesian closed category and closed monoidal category.
(Co)reflective subcategories. This is related to Pece's answer. Often you have a category $C$ and a full subcategory $D$, and also not only the inclusion functor $D \to C$ but a "$D$-ification" functor $C \to D$; this is precisely a left adjoint to the inclusion $D \to C$, and in this situation we say that $D$ is a reflective subcategory of $C$. The idea is that being in $D$ is a property of an object in $C$ and there's some universal way to force an object to have that property.
Archetypal examples include the inclusion $\text{Ab} \to \text{Grp}$ (whose left adjoint is Abelianization), the inclusion $\text{Sh} \to \text{Psh}$ from sheaves to presheaves (whose left adjoint is sheafification), the inclusion $\text{Haus} \to \text{Top}$ from Hausdorff spaces to spaces (whose left adjoint is Hausdorffification), the inclusion $\text{CHaus} \to \text{Top}$ from compact Hausdorff spaces to spaces (whose left adjoint is Stone-Čech compactification), etc. Of course there is a dual notion involving right adjoints but this seems to occur less often.
Galois connections. This case is in principle quite specialized but in practice occurs surprisingly often. It turns out that a relation $R : X \times Y \to 2$ between two sets induces a (contravariant) adjunction between the posets $2^X, 2^Y$ of subsets of $X$ and $Y$. An adjunction between two posets induces closure operators on each poset, and it's usually interesting to ask what the closed subsets are. Three important examples: the relation "$g \in G$ fixes $\ell \in L$" induces the adjunction between subsets of a Galois group and subsets of a field extension that is the subject of the fundamental theorem of Galois theory. The relation "$f \in k[X]$ vanishes on $x \in X$" induces the adjunction between subsets of a variety and subsets of its field of functions that is the subject of the Nullstellensatz. The relation "statement $S$ is true in model $M$" induces the adjunction between subsets of statements and subsets of models of some theory $T$ that is the subject of Gödel's completeness theorem (Lawvere's slogan: "syntax is adjoint to semantics").
This example can be understood as a very special case of a very general kind of tensor-hom adjunction involving enriched bimodules.
My personal favorite description uses profunctors.
If $\mathcal A$ and $\mathcal B$ are categories, a profunctor $F:\mathcal A\not\to\mathcal B$ "from $\mathcal A$ to $\mathcal B$" is defined as a functor $F:\mathcal A^{op}\times\mathcal B\to\mathcal{Set}$. The bigger category that disjointly contains $\mathcal A$ and $\mathcal B$ and the sets $F(a,b)$ are considered as collections of further arrows ('heteromorphisms') from $a\in\mathcal A$ to $b\in\mathcal B\ $ is called the collage of a profunctor, and uniquely determines that.
Now, any functor $T:\mathcal A\to\mathcal B$ determines two profunctors in a natural way:
$T_*:=\hom_\mathcal B(T-,\,-)$ from $\mathcal A$ to $\mathcal B\ $ and $\ T^*:=\hom_\mathcal B(-,\,T-)$ from $\mathcal B$ to $\mathcal A$.
By definition, functor $L$ is left adjoint to $R\ $ iff $\ L_*\cong R^*$, and that means that they have a common profunctor.
Moreover, we have that a profunctor $F$ is functorial (isomorphic to $L_*$ for some functor $L$) iff $\mathcal B$ is a reflective subcategory of the collage (each object of $\mathcal A$ has a reflection in $\mathcal B$), and dually, $F$ is cofunctorial (isomorphic to some $R^*$) iff $\mathcal A$ is a coreflective subcategory.
Examples:
The heteromorphisms of the profunctor $\mathcal{Set}\not\to\mathcal{Grp}$ that belongs to the free/forgetful adjunction of groups are just the functions from a set to (the underlying set of) a group.
[Actually, this is exactly $U^*$ for the forgetful functor $U:\mathcal{Grp}\to\mathcal{Set}$, but the point is that each set has a reflection among groups (the free group on that set) in this collage.]For a noncommutative ring $R$ and a fixed left $R$-module $U$ consider the profunctor $\mathcal{Mod}_R\not\to\mathcal{Ab}$ with $R$-bilinear maps $M\times U\to A$ as heteromorphisms $M\to A$.
This is the profunctor for the tensor-hom adjunction.Let $\mathcal{Mat}_k$ denote the category whose objects are natural numbers and morphisms $n\to m$ are $n\times m$ matrices with entries of a field $k$, and let $\mathcal{Vect}_k$ be the category of vector spaces over $k$.
Define the heteromorphisms $n\to V$ as the $n$-tuples of elements of $V$, i.e. consider the profunctor $F(n,V)=V^n$, compositions (i.e. actions of $F$ on arrows) are straightforward.
This profunctor defines a full embedding $\mathcal{Mat}_k\hookrightarrow\mathcal{Vect}_k$, where a heteromorphism $(v_1,v_2,\dots,v_n):n\to V$ is a reflection and coreflection arrow at once iff it is a basis, whence $V$ can be any $n$ dimensional vector space, and the reflection/coreflection properties guarantuee the faithfulness.Let $\mathcal{Rel}:\mathcal{Set}\not\to\mathcal{Set}^{op}$ be the profunctor with binary relations $r\subseteq A\times B$ as heteromorphisms $A\to B$. If we write composition in $\mathcal{Set}$ from left to right, and in $\mathcal{Set}^{op}$ from right left and use infix notation for the relation, then the compositions can be neatly defined as $$ A'\overset{f}\to A \overset{r}- B\overset{g}\leftarrow B' \\ a'\,(frg)\,b'\,\iff\, (a')f\,r\,g(b')\,.$$ This is the profunctor of the self-adjunction of the contravariant powerset functor.
In general, I suspect an adjunction whenever a functor preserves limits of colimits. I then try to find the adjoint or to apply Freyd's theorem. Of course, sometimes I fail but as "adjoints are everywhere" I also often win.
But there is a particular case when I find it easy to see adjunctions. Take a category $\mathsf C$ and a full subcategory $\mathsf D \stackrel i\hookrightarrow \mathsf C$. Then the inclusion $i$ has a left adjoint if and only if every arrow $c \to d = i(d)$, with $c \in \mathsf C$ and $d \in \mathsf D$, uniquely factorizes through some $\tilde c \in \mathsf D$. $$ \require{AMScd} \begin{CD} c @>>> \tilde c \\ @| @VV\exists! V \\ c @>>>d \end{CD} $$ The left adjoint is of course $c \mapsto \tilde c$. For example, the inclusion of presheaves into sheaves admits the associated sheaf functor as left adjoint, the inclusion of locally ringed spaces admits the $\operatorname{Spec} \circ \Gamma$ (composition of the global sections functor and the spectrum functor) as left adjoint, the inclusion of the $R$-modules ($R$ fixed ring) in which all elements of $S$ are invertible coefficients ($S$ fiwed multiplicative subset of $R$) into the $R$-modules admits the localization functor $S^{-1}\!\!-$ as left adjoint, etc.
(Remark that you do not need $i$ to be a fully faithful inclusion ; I just find it more easy to check the unique factorization when it is the case.)