The director provides the vision. Many directors also possess the necessary skills related to the industry, such as writing and programming, in order to better guide the team. Case in point, Junichi Masuda is probably better known as the composer behind most of the Pokémon game music, but he also has experience/knowledge in directing and programming. It's as the word means: the director directs the team and points them in the right direction (yes, that's a lot of directs).
However, the director will oftentimes also report to the "producer." A game producer controls the funding for the game. As the producer manages the funds, they have a say in the direction of the game, too. In many cases, for the sake of streamlining, you'll also see someone being credited as both the director and producer. For example, Junichi Masuda was the director for Pokémon X and Y, plus he also served as one of the game's producers.
You can see it like:
The producer(s) lead the overall game (he who has the money gets to do the talking!). The game director leads the team of developers (programmers, writers, etc.).