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.