In terms of the clubhead, one with a higher MOI will be more resistant to twisting than one with a lower moment of inertia. Why does that matter? For example, your swing is a little off and you hit the ball on the toe of the clubhead. That will cause the toe of the clubhead to push backward just a smidge (opening the clubface). A high-MOI clubhead will twist less on such a mis-hit, which can minimize the effects of the mis-hit on the flight of the golf ball.
The way that manufacturers boost a club's MOI is by playing with the weighting properties; any object will increase in MOI as more of its weight is moved outward, around its perimeter. (This is one reason perimeter weighting led to the game-improvement club category, and a reason why manufacturers oftet tout weight plugs around the perimeters of clubheads. Clubs that are described as "more forgiving" usually have high MOIs.)
MOI is a physical property that can be expressed as a numerical value. Such measurements are starting to be seen more often in golf manufacturers' advertising and websites, although they can't yet be called common.
For more on MOI, including more technical descriptions, see the FAQ, "What is moment of inertia?"
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