Penalty doubles  

Definitions

Doubling for Penalties

(this double means you don't want to play you want to penalise the opposition)
You can double a contract if your opponents arrive at a final contract that either you or your partner think is just too high. For example, if the opponents bid 7NT (notrump), meaning that they must take all 13 tricks, and you're on lead with an ace, you know that they can't take all 13 tricks. Penalty doubles are made by writing "X" on the bidding pad and are usually made near or at the end of an auction, there must be at least three Passes "P" after a double before the auction can close.

You can  “Double” for penalties when it’s your turn to bid.  If you defeat the doubled contract, you get at least double your normal penalty score for the under tricks however if the opponents make the doubled contract, they get double their trick score plus 50 for the insult. 

Redoubling for Penalties 

(this double means you don't want to play you want to penalise the opposition) If your opponents have doubled you and you or your partner think that you will actually make your contract, when its your turn bid you can re-double ("I bet we can make it!") their double. The penalties are then  much larger than if the contract had been just doubled. A redouble is written "XX" on the bidding pad.