There are different requirements for a takeout double after your opponents have opened the bidding, depending on the strength of your hand.
A takeout double can get your side into the bidding when your hand is not suitable for an overcall (ie. no good-quality 5-card suit).

12-15 HCP

With this strength, shape is very important.  First, you should have no more than 2 cards in the opposition’s suit - with a singleton or void in their suit, your hand can be a bit weaker, especially if you’re not vulnerable.
Second, you must have at least 3 cards in the other 3 suits.  In other words, you have a fit with any 5-card suit your partner bids in reply.  If the opponents’ suit is a major, you should have 4 cards in the other major; if a minor, then 4-3 in the majors is OK.

If your hand is not the right shape (3+ cards in their suit; fewer than 3 in any unbid suit) you should PASS!  If you double with the wrong shape, partner is sure to call the suit you can’t support!  Don’t worry about passing – if the next player passes too, your partner can make a Balancing Double to reopen the bidding.

16+ HCP

With a stronger hand you can double for takeout on any shape.  When you double, then bid again after partner’s minimum response, you’re showing this stronger type of hand.
With 15-18 HCP and a hand suitable for a 1NT opening, bid 1NT instead of making a takeout double if you have a double stopper and at least 4 cards in their suit.
If you double for takeout then rebid NT you are showing a very strong hand – 19+HCP.

Responding to a Takeout Double

Partner’s takeout double says:  Please bid your best suit – I have an opening hand.  No matter how weak your hand, you must answer the double unless the other opponent has bid in the meantime.
You should bid at the appropriate level, depending on your hand strength.
0-9 HCP – bid your longest suit at the lowest level (minimum response)
6-9 HCP – bid 1NT with no long suit and at least 1 stopper in the opponent’s suit.
10-12 HCP – bid your long suit or NT with stoppers, as above, but jump one level.
13+ HCP – bid game in your long suit or bid 3NTs if you’re sure you have their suit well stopped - Axx is just not good enough!

Remember that partner has values in the other three suits – that’s why they’ve doubled.  If they have 16+ HCP they will probably bid again - otherwise they will pass any minimum bid you make in reply.

Passing Partners Double!!

There are few exceptions to the rule that you must answer your partner’s double. 
If the doubled bid is at the 3-level or higher  (eg. a pre-empt) and you have strength and length in their suit, you can convert partner’s takeout double to a penalty double by passing.  Your partner’s values will be in the other suits – if you can be sure of making at least a couple of trump tricks yourself, you may decide to leave the double in.  But it’s very dangerous to do this at the 2-level, and risky even at the 3-level.