In trying to think of an intuitive answer to [a question on adjoints](https://mathoverflow.net/questions/6376/why-forgetful-functors-usually-have-left-adjoint), 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](http://ncatlab.org/nlab/show/adjoint+functor) (and any good answers will be added there), I've [worked](http://www.math.ntnu.no/~stacey/Research/Papers/hopf.html) [with](http://www.math.ntnu.no/~stacey/Research/Preprints/smthcat.html) [them](http://www.math.ntnu.no/~stacey/Research/Papers/deloop.html), 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.