In the UK, the director of a business is basically in charge. It's similar to CEO in the USA.
Note the use of "the", which implies a singular, versus an "a", which would imply multiple directors.
It's also a legal thing, I am joint director (with my Wife) of my own company, our company has 2 employees (me and my Wife!). Under UK law, when you set up a business (Public Limited Company - PLC) you need to declare at least 1 person as a director and 1 as company secretary. Under the law they then have certain responsibilities (such as filing tax returns etc.).
However, when a larger company is involved (such as Grunnings), although we have no idea how large it is, the term 'director' may not mean director in a legal sense. I suspect that in this case it simply denotes that he's higher up the managerial chain than some.
In the United States, however, the term director is typically used to describe a senior middle-level manager, just short of "managing director".
Other terms that can also mean similar things are:
Manager, CEO, MD (Managing Director) etc.