Reopening Takeout Doubles

Definition

In contract bridge, a takeout double is a low-level conventional call of "Double" "X" after an opponent's bid, this is a request for the doublers partner to bid his best of the unbid suits.
The most common takeout double is after an opponent's opening bid of one of a suit where the double shows a hand with opening values and a shortage in the suit of the opening bidder.

Takeout Doubles (showing an opening hand and not for penalties)

A double by the defense at their first bid is usually intended to be taken out by their partner. Partner does this by bidding their own suits, it is not usually for penalties (punishing the opposition). Even if you have no points it is a forcing bid and the doublers partner is required to bid their cheapest four card suit. A takeout double can be made after the opponents have opened the bidding. It shows 13+ TP and at least 3 cards in all of the unbid suits. Partner must reply even if they have no points in their hand (in that case they should bid their cheapest 4 card suit). 
Some partnerships have the understanding that when responding to a takeout Double, a minimum bid indicates less than 8 HCP, and a jump bid indicates 8 or more HCP
Double is written by making an “X” on the bidding pad.

Takeout Doubles in later rounds of Bidding

Double can be for reopening takeout in later rounds of bidding. It shows support for the unbid suits and sometimes extra values. Generally any low-level double after opponents have bid a suit (especially if opponents have found their fit) and partner has previously passed.

Reopening Takeout Doubles

Opener can double for takeout in the second round of bidding, when the opponents have overcalled and partner had either bid or passed. As with a normal takeout double, it shows support for the unbid suits. In this situation your double is showing good spades and/or diamonds, asking partner to choose the best contract.