I like Wikipedia's motivation for an adjoint functor as a [formulaic solution to an optimization problem][1] (though I'm biased, because I helped write it).  Here's a digest version of the discussion:

An adjoint functor is a way of giving the *most efficient solution* to some problem via a method which is *formulaic* ... For example, in ring theory, the *most efficient* way to turn a *rng* (like a ring with no identity) into a *ring* is to adjoin an element '1' to the rng, adjoin no unnecessary extra elements (we will need to have *r*+1 for each *r* in the ring, clearly), and impose no relations in the newly formed ring that are not forced by axioms.  Moreover, this construction is *formulaic* in the sense that it works in essentially the same way for any rng.  

The description of this construction as "most efficient" means "satisfies a universal property", in this case an initial property, that it is "formulaic" really means that it's functorial, making it an adjoint functor.

In this asymmetrc interpretation, the theorem (if you [define adjoints via universal morphisms][2]) that adjoint functors occur in pairs as the following intuitive meaning:

The notion that *F* is the *most efficient solution* to the problem posed by *G* is, in a certain rigorous sense, equivalent to the notion that *G* poses the *most difficult problem* which *F* solves.**


  [1]: http://en.wikipedia.org/wiki/Adjoint_functors#Adjoint_functors_as_formulaic_solutions_to_optimization_problems
  [2]: http://en.wikipedia.org/wiki/Adjoint_functors#Definitions_via_universal_morphisms