Let me expand a bit on @Sagacious .
The director is the person whose artistic vision ties all the pieces of the movie together – choice of scenes, music, dialog and how it is delivered, plot tension, and so on and so forth.
The producer pulled together the operational aspects – budget, location, (usually) the actors, and lots of other details that make movies possible. But the director is the one responsible for ‘telling the story’ using the factors I mentioned above.
In short, a producer deals with money and facts, and the director figures out how to effectively tell the story.
To answer your question, @LostInParadise – the people voting on Best Director are really making an assessment – what director best told the story in the film?