They represent the shareholders. They usually have the power to hire/fire the CEO and other CxO level management. They run "the company on behalf of its members

They represent the shareholders. They usually have the power to hire/fire the CEO and other CxO level management. They run "the company on behalf of its members". They're expected to set the company's strategy, are responsible for the company's regulatory compliance, ensure that the company is properly reporting its performance to its shareholders, that it doesn't trade while insolvent, etc. There are "executive directors" that have management responsibility, eg the CEO and "non-executive directors" that do not. In the US, often the CEO and the President of the board are the same person, but that's less common elsewhere and is considered a flaw in corporate governance. There is also the concept of "independence" where a director is independent of related interests, eg being also related to other corporations that have major business as a supplier or customer etc.