A nonprofit board member's Duty of Loyalty is one of the three duties of which members are held responsible and accountable severally and collectively particularly by courts but also by all others, including board members.
Traditionally, the go-to actionable definition of duty of loyalty is associated with Conflicts of Interest. When a member puts their own, often financial, interests ahead of the organization, they are failing their duty of loyalty. Not telling the board (non-disclosure) of a possible or substantive conflict is included in this fail. A simple example: a new roof is needed. A board member offers that he knows a firm that will do the job for a "reasonable" price in a "reasonable" way. Recognize of course that all bidders being equal, a board member's word carries a bit further in the group's decision just because he is a board member. The member gains added authority by being a board member. What the board member does not mention is that the roofer in question is a relative (e.g. sister or sister-in-law) which in turn provides a benefit, albeit not specifically direct, to the board member. Indirect and direct conflicts must be declared to the board so the board is fully informed in its decision-making. The roofing job maybe perfectly fine and price perfectly competitive but by declaring the relationship, the board can determine if their decision in some way lessens the benefit to the organization. The organization benefit, for a duty of loyalty, must always remain the primary objective of every decision.
Now, if you would like to take the concepts of duty of loyalty to an even higher level, check--out Forbes Advisory Council member Christopher Washington's thoughts on this subject where he offers
....the idea in the manner expressed by Josiah Royce, in his book The Philosophy of Loyalty. Royce defined loyalty as "the willing and practical and thorough going devotion of a person to a cause."
My takeaway is that a simple legalistic view of Duty of Loyalty produces just that, playing by the rules. But cultivating a deep loyalty to the organization, particularly in board members, results in a proactive set of actions that produce uber positive outcomes for the organization. Rather than just believing that mission development is enough, take steps to build a loyal fan base in the board - one that in itself results in action in the best interest of the nonprofit.