TITLE: What is an intuitive view of adjoints? (version 1: category theory)
QUESTION [71 upvotes]: In trying to think of an intuitive answer to a question on adjoints, I realised that I didn't have a nice conceptual understanding of what an adjoint pair actually is.
I know the definition (several of them), I've read the nlab page (and any good answers will be added there), I've worked with them, I've found examples of functors with and without adjoints, but I couldn't explain what an adjunction is to a five-year-old, the man on the Clapham omnibus, or even an advanced undergraduate.
So how should I intuitively think of adjunctions?
For more background: I'm a topologist by trade who's been learning category theory recently (and, for the most part, enjoying it) but haven't truly internalised it yet.  I'm fully convinced of the value of adjunctions, but haven't the same intuition into them as I do for, say, the uniqueness of ordinary cohomology.

REPLY [5 votes]: Adjoints can be viewed intuitively as expressing conceptual duality between mathematical notions.
Examples along these lines include $${\sf Free}\dashv {\sf Forgetful},$$ $$\varinjlim\dashv\Delta\dashv\varprojlim,$$ where $\Delta:\mathcal{C}\to\mathcal{C}^\mathcal{D}$ is the diagonal functor on a (co-)complete category $\mathcal{C}$, $$\exists\dashv*\dashv\forall,$$ where $*:{\sf Form}(\overline x)\to{\sf Form}(\overline x,y)$ is the functor adjoining a variable $y$ not appearing in $\overline x=(x_0\dots,x_n)$ which we quantify existensially or universally over, $$\lceil x\rceil\dashv i\dashv\lfloor{x}\rfloor$$ as noted by user1421 in his answer, where $i:\mathbb{Z}\to\mathbb{R}$ is the standard inclusion - there are many more that belong on this list.
In essence, if two notions are related to eachother closely enough that giving a definition of one notion (in the presence of sufficient ambient structure) defines the other notion completely, they can be expressed as a (possibly $\infty$-) adjoint pair of functors. Conversely, when we see that two notions can be expressed as an adjoint pair of functors it means we can think about one by thinking about the other plus some canonical additional structure.
In the above cases, one of the two notions is significantly simpler than the other -- forgetful functors are simpler than free object functors, diagonal functors are simpler than limit and colimit functors, adjoining a variable is simpler than existentially or universally quantifying, etc. This means we can understand complex notions by understanding simple ones and making them functorial, then considering their left or right adjoint. Although there may be another way to construct whatever notion we end up with, any two right (or left) adjoints to the same functor will be naturally isomorphic, so we've gotten at the definition safely and using tools that are very well understood and easy to manipulate.