Weak interference arises when an interference situation occurs that is not serious enough to warrant a penalty, but significant enough to require umpire intervention to maintain fair play. The term "weak interference" does not appear in the rule book, but includes different situations that arise from time-to-time that most commonly would be referred to as "accidents" or "freak plays."
The most common example is the batter calling time and (without the umpire grant) causes the pitcher to balk.
Another common/uncommon example is the catcher hitting the batter on a return throw:
In a weak interference call the ball is dead and the runners return.
Here is another well-known example of weak interference with pitcher Randy Johnson:
In this case the bird causes a weak interference call. A "no pitch" is called, the ball is dead, the bird is dead, and runners return to their bases.
Another common case is the batter "interfering" with the catcher on his backswing. Under FED rules, the batter is out for interference. Under OBR it is weak interference, the ball is dead and runners return.