Suppose the players are Anne, Bill, Charlize, David, Elizabeth, and Fred.
Anne rolls the dice, moves her counter and lands on Fred's property. Fred fails to notice.
The players following Anne are Bill, Charlize, David, etc. Bill is the first player following Anne, and Charlize is the second. So Fred must notice and request rent before Charlize rolls.
A specific scenario:
The first following player, Bill, now rolls the dice and moves his counter. Any transactions are processed as necessary.
The second following player, Charlize, now reaches for the dice as Fred notices Anne's counter and request payment. As the second following player, Charlize, has not yet thrown the dice, the request by Fred is in order and must be honoured.
The question is now asked: "why the second following instead of first following player?"
Simply, to prevent a pair of players from sitting down and guaranteeing that the one in elder position wins. Under a "first following" rule, if Alfred and Betty sit down beside each other, with Alfred rolling first, then Betty simply must be very fast at rolling the dice to ensure that Alfred never pay rent.
Next game, Alfred and Betty switch places and Betty is guaranteed to never pay rent.
And so on. This is called "The Bum's Rush".
There are also more subtle ways to play it, where the younger hand rolls very quickly only at strategic moments in the game. Same effect, with only a very slight reduction in effectiveness.