The re-opening double is an essential part of bidding. To start, it goes like this. You open the bidding, your LHO makes an overcall at the one or two level, and your partner and RHO passes. It is now back to you. Since you all play negative doubles through at least 2S, and most times to a higher level, partner could not double the overcall as a penalty double. So, partner passed for one of two reasons: they are weak and have nothing to say, or they would have liked to make a penalty double but could not because of the negative double convention you are playing. How do you handle the situation?
The first, and probably the most important rule, is this: whenever you hold 2 or fewer cards in the overcall suit, assume partner has values but could not act because of the negative double convention. The converse is also true: whenever you hold 3 or more of the overcall suit, assume partner is weak and only act on the merits of your own hand.
When the rule applies, re-open the bidding. Whenever you have support for the unbid suits and have the 2 ½ quick tricks of a normal opening bid, re-open with a takeout double. Without this hand structure, either rebid your suit or bid your second suit.
When you re-open, you are protecting partner when they have a hand with values, but no convenient bid to make. Consider this hand with you as west and partner as east:
The bidding has gone: West 1S; North 2C; East & South Pass; West Double; All Pass
2C doubled will go down at least 2 and you cannot make game. East would lead the spade J, West would play A, K, and another spade that East trumps. East-West will get at least two more club tricks and their two aces. North does not have a good enough suit to overcall, and would be penalized for this bid against such as you and your partner.
Partner has values and could bid 2D over the 2C overcall, but the singleton spade and excellent club holding suggest a misfit and maybe defense would be better. Partner knows you will be short in clubs and will reopen the bidding, and if you re-open with a double they will pass and defend 2C doubled.
The two thoughts to keep in mind in all hands of this type are:
~as responder, when you are short in opener’s suit, have length and honors in the overcall suit, and no really good bid to make, pass and await developments
~as opener, always reopen when you are short in the overcall suit, and with a double when holding support for the unbid suits and expected defense strength.