They have an extremely vague "duty of care", i.e. to run the company in a manner in accordance with the wishes of the entities they serve as representatives of. Beyond that, they don't have any specific obligation; shareholders can, at least in principle, have many different wishes, and so elected board members can represent a variety of positions, from an aggressive profits-uber-alles position to some kind of safeguard-the-brand-reputation-for-generations viewpoint.
The main enforceable obligation is a negative one, to not actively do things that benefit themselves at the expense of the corporation they oversee, e.g. by making decisions primarily designed to enrich themselves personally. Almost anything that isn't active wrongdoing is defensible though; if a board member thinks in good faith that doing X would enhance the goodwill towards the Apple brand, and in good faith thought that prioritizing brand goodwill was the best long-term strategy, it would be fine to undertake a short/medium-term money-losing course of action to pursue the strategy. Courts generally defer to board elections to resolve those kinds of disputes over strategy, since courts are very bad at predicting whether a given strategy is actually in a particular entity's long-term interests.
Insurance these days is more often directed at government regulations than shareholder lawsuits; board members have various possibilities for personal liability if their company is doing illegal things on their watch.
The main enforceable obligation is a negative one, to not actively do things that benefit themselves at the expense of the corporation they oversee, e.g. by making decisions primarily designed to enrich themselves personally. Almost anything that isn't active wrongdoing is defensible though; if a board member thinks in good faith that doing X would enhance the goodwill towards the Apple brand, and in good faith thought that prioritizing brand goodwill was the best long-term strategy, it would be fine to undertake a short/medium-term money-losing course of action to pursue the strategy. Courts generally defer to board elections to resolve those kinds of disputes over strategy, since courts are very bad at predicting whether a given strategy is actually in a particular entity's long-term interests.
Insurance these days is more often directed at government regulations than shareholder lawsuits; board members have various possibilities for personal liability if their company is doing illegal things on their watch.