It really depends. As a director you should NOT be liable to the ordinary creditors of the company (such as a customer, or landlord) unless you did something wrong or were personally involved.
You could be liable to shareholders if your conduct was negligent and asa result the company or shareholders suffered. If there is a real serious problem, as there is at Enron, the directors will get caught up in it as the creditors will likely try to go after anyone in sight, even if they are not ultimately liable.
As far as the insuance goes, that depends on the nature of the matter and if the insurance is broad enough. If a customer is injured by a defective product the General Liability carrier will cover the company's liablity TO THE EXTENT OF THE POLICY and thus protect a director. If the board does a stupid or dishonest deal, or is asleep at the switch, the other shareholders may sue. Ordinary general liability insurance would not protect directors, only director's and officers' liability would.