Responsive Double

The responsive double is so named because it's a double in response to partner's takeout double
after the opponents have bid and raised a suit. Sounds complicated? Here it is in bidding format:

(1H) - X - (2H) - X 

The responsive double is based on the principle that a penalty double here is usually unrewarding.
Therefore double shows scattered values with at least 6 points and interest in locating a fit.
After a major suit opening, a responsive double shows both minor. 
After a minor suit opening, a responsive double shows both major (4-4).

This is the easy way to use it. Here is how I use the system:

After (1S) - X - (2S) there can be a problem what 3H now means. Is it to play, or is it invitational?
To solve this problem I use 3H directly as a playing bid (6-9 HCP). If you wish to make an invitational bid in heart, you first have to double (responsive), and then bid 3H over partners 3C/3D.

(1S) - X - (2S) - X (responsive double)
(P) - 3C/3D - (P) - 3H : Invitational in heart.

If the opponents bid 3S before you have the chance to bid 3H, you must double to show an invitational hand in heart. 

You seldom double 1S if you don't have 4 cards heart or a strong hand. But after 1H you may double with only 3 card spade since spades is the highest suit. 
After (1H) - X - (2H) I use 3S as an invitational bid with 5+Spades. If you have an invitational bid with only 4 card spades you first has to double (responsive) and then bid 3S over partners 3C/3D.

Note that a double by advancer is responsive only when the opponents have opened a one-bid and
raised it. Some players also use the responsive double when partner has made a simple overcall in a
suit. Modern Bridge Conventions by Bill Root and Richard Pavlicek gives an excellent list of
bidding situations in which double is NOT responsive:

(1C) - X - (1S) - X
RHO has bid a new suit. (Optional treatment)

(1C) - 2S - (3C) - X
Partner has made a jump overcall.

(2H) - X - (3H) - X
LHO has opened with a preempt

(1D) - 1NT- (2D) - X
Partner has overcalled in no-trumps.

How high to play responsive doubles is a matter of partnership agreement.