An operations director flits between preparing budgets, handling business logistics and ensuring employees are being suitably supervised. Therefore they need to be able to multitask in various areas, and be able to communicate well. Organisation skills are also essential as is having some top-drawer analytical and problem-solving skills.
Because they are involved in so many different departments, knowledge of different computer programs is needed, or at least the ability to learn new software.
Communication is another key skill to have. Not only do you need to identify where things aren't working, you need to be able to communicate this, plus give instructions on how to make things better, (for example involve holding training seminars or writing manuals). Plus you need to be able to communicate with executive management, and be prepared to argue your case.
Judgement is also an important skill to have, not only for handling situations but also for making decisions across many departments.
Staff development is a big part of an operations director role, so you need to be able to inspire energy and creativity. Obviously as a figure that deals with problems across many departments, you need to be professional at all times.