What powers and responsibilities do for-profit corporate board members have?

Directors owe the company fiduciary duties: loyalty and due care. More details are found in the relevant state's corporation law. Because boards act only collectively, individual directors technically have almost no power of their own, unless they get special protective provisions creating a veto right. Directors can't do much except make sure their opinions are heard at meetings before votes are taken. However, the practical effect of a single director vote can vary widely depending on the composition of the board. Two investors, two founders, and all of a sudden the fifth director might seem to be the only one with power. But its important to focus on getting good directors. Finally, an alternative to having the VC partner on the board is asking the VC to instead nominate an independent director to fill that seat. While a founder might expect the independent director to be sympathetic to investor interests, it's important to keep in mind that directors are individually liable for their breaches of fiduciary duties. That is supposed to keep the independent director independent, and if the choice is well-made, it will be a person whose opinion is respected and therefore valuable, regardless of who nominated the director. Put differently, once they're on the board, directors are legally bound to serve the company and not a subset of shareholders.