I like to think of an adjoint functor as a [formulaic solution to an optimization problem][1].  It's "formulaic" because it defines a functor, and it solves an optimization problem because of the universal properties exhibited by adjoints.  Read the link for a detailed explanation of this.

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