"A type for context-dependent computation"
Alternatively, a better "pithy description" for monads might be a 'type for output impurity', in which case then the pithy description for comonads is a 'type for input impurity'.
(If you are interested in comonads, some more introduction is given in some talks slides of mine: http://www.cl.cam.ac.uk/~dao29/talks/comonads-and-codo-talk-dorchard-2011.pdf)