You will need
  • - knowledge of the etiquette and rules of politeness;
  • - a sense of self dignity.
Whatever the reason for communication, one should not forget that the Director is the same person as any other. And ask you may not more than that stipulated in the terms of your responsibilities, corporate standards and applicable law. Latest corporate culture may contradict and indicate the priority in this case for the law.
If the initiator of the meeting - you try to briefly but succinctly explain the essence of the question, it is possible, in your opinion, solutions to problems, if we are talking about her proposals and arguments in their support.
Otherwise, assume that should be the boss of you. Clearly, briefly and succinctly answer questions, be prepared to make proposals, make necessary refinements and adjustments.
Never forget that time Director of the limited, he has many important tasks that require his participation, he thinks on the job in the first place, and their subordinates, starting as a cleaner and finishing with his right hand, waiting for the same.
How to appeal to the boss, depends on your company standard. Even if it's practiced dealing with the management of "you" (but "you" more often), the first time to treat "you" to a unfamiliar person more than not ever, and to go to "you" at any time, especially if it offered itself the other side.
In any case, appropriate observance of the norms of courtesy on your part and demand the same from the interlocutor. You have nothing to fear: those who can, if necessary, show the teeth (but don't abuse this skill), respect in the soul of even the most rabid louts. In an extreme case, the best solution will be to find another company with more sane Director.