My oversimplification: Board members usually come in four flavors:

Company officers such as the CEO, CFO, executive VPs and Senior VPs (vanilla) - they have the most intimate knowledge of how the firm does what it does

Officers of related companies that are not direct competitors (chocolate) - they also contribute generally related expertise in how the company functions best

Non-officer holders of a significant number of shares of the company's stock (strawberry) - they're here because they have "skin in the game" (and could probably vote enough shares to get themselves on the board anyway).

"Independent" board members - These are chosen based on their individual reputations, but generally do not have a specific relation to the company (pistachio). Often, boards will require themselves to have a certain proportion of members that are "independent" and typically one of this group would chair a compensation committee (or other committees where conflicts of interest would most likely otherwise develop). This will tend to pacify shareholders who might otherwise question the judgement of the board in general.

Typically a company will have a Nominating Committe (of board members) whose job it is to submit nominations for board members whose terms are expiring, to be voted on at the annual shareholders meeting. Sometimes a group of shareholders will offer alternatives to the nominations if they are dissatisfied with the choices offered by the nominating committee - this can often lead to much turmoil in the run-up to a meeting, but it keeps the shareholders in charge of what the makeup of the board is.

So theoretically, if you and your friends had a controlling interest of shares in a company between you, you could get yourself voted onto the board even if the other members of the board were opposed to it. Doesn't happen often, but you will see it every now and then.