Negative doubles occur when partner opens the bidding with a natural suit bid, and opponents make a natural overcall at any level. Most people nowadays play negative doubles, but it is surprising how many of those are unaware of the basics.
A negative double shows 6+ points at the one level, about 8+ at the two level, and 10+ at higher levels. There is no upper limit to the strength of a negative double.
If in doubt as to whether you have the strength for a negative double, look at your shape, and consider what might happen if you do not make a negative double. For example, it starts 1-2, and you have a seven count with 4432 shape. This is a clear negative double. Although a little light in values, you have nearly the strength, and would much prefer to let partner choose a suit than yourself.
If partner opens a minor and they bid a major, or partner opens a major and they bid a minor, then the double usually shows four cards in the unbid major. For example, 1-1-X would show exactly four spades, whilst bidding 1 here would guarantee five. This is because you can freely bid 1 without showing extra values. 1-1-X shows at least four hearts, though it may be more hearts, but only if the hand is too weak to bid 2 (eg Ax Q109xxx xxx xx - provided that the bidding does not get too high you will later bid hearts to show a weak hand with a long heart suit). Similarly it denies a strong hand with five diamonds and four hearts - with that you would start off by bidding 2.
1-1-X shows 4-4 in the majors. 1-2-X implies at least one four card major, though may well not have both. 1-1-X is in principle showing the minors but may well just be conveying some values.
In each case you can bend the rules if you are likely to be able to survive later in the auction. If it starts 1-1, you can afford to double with a weak hand with six diamonds, intending to convert clubs to diamonds. If it starts 1-2, you can afford to double without four hearts if you have three spades - you can later convert back to spades, since if partner has four hearts, he will also have five spades, given that he opened 1. That is not to say you should bend the rules willy nilly - only do so for a good reason!
The higher the negative double, the less specific it is about shape. Generally a strong hand will never make a negative double with a 5 card suit.
A negative double should never be made with a fit for partner's major, and usually not with a fit for partner's minor at higher levels.
Responding to a negative double:
Most responses are non forcing. For example 1-2-X-P: here 2, 2, 2, 2NT, 3 and 3 would all be non forcing - pretend that partner has responded 1, and you won't go far wrong.
Continuing after a negative double:
New suits after earlier making a negative double show a weak hand. To introduce a new suit you will obviously want some length in the suit, and with a good hand you would simply have bid the suit straight away. For example, 1-1-X-3, P-P-4 will typically be a weak hand with four hearts and six clubs.
Passing rather than doubling:
If you pass an overcall that opponents make, it says that either you haven't got the necessary stuff to take an action, or that you have penalty double of their suit. Several points to note:
1. 1-1: Do not pass with eg xxx KQx xxx Qxxx. You must bid 1NT on this - otherwise partner will never believe you have these values later on.
2. Pass is not forcing. Instead, the opener should look at his length in their suit to decide whether to reopen with a double. With a doubleton or fewer, he should generally reopen at the one or two level, irrespective of his strength, whilst with a singleton, he should reopen at the three level. If occasionally you pass them out when you have three cards in the suit, and your partner has a penalty double of it, you will be amazed how often it gets you a good score - they probably have a far better contract elsewhere.
3. When deciding to pass, think what action you would take if partner reopens, and on this basis decide whether you would have been better taking some action in the first place. For example, after 1-2-P-P, X-P what would you bid with a 4432 7 count? Any of three suits could be right, and you have no idea whether 8 tricks is the limit of the hand, or whether game might be on.
4. If you pass and later double, this is showing a penalty double of the suit you passed over.
5. If you reopen with a double and partner passes, this does not necessarily show values - it merely shows that partner felt that defending their suit was likely to be your best result. For example, if it starts 1-2-P-P, X-P-P-2, X, don't be surprised if this makes if you have doubled this with a minimum opening hand - partner may only have defensive tricks in clubs.
Editor's note: The above is based on negative doubles where if instead you bid a suit it would be forcing (eg 1-1-1 is forcing - though not necessarily strong). It is absolutely clear that this is better than playing non-forcing bids, and having to start with a double with strong hands with a good suit. There are different styles available here as to shape (some genuinely worth playing), but we stick with the commonest.