Quoting @Meadmaker:
A quad is a four player round robin.
If there are twelve people in a tournament, to play in a quad format, you take the 12 players, and put them in order of their ratings. Then, the top 4 play a round robin against each other, the next four play a round robin against each other, and the four lowest play a round robin against each other. Each round robin requires three rounds of play, for a total of six games.
The quad format does not produce an overall winner. It only produces a winner within each quad. It's a good format for a schedule with time available for only three rounds, and low prizes. Every player gets to play three rounds of Chess against players that are roughly equal in rank, and usually has at least a passing shot at first place within his group.
I do not know what a rapid quad tournament is... could someone explain it