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.
