It can sometimes be confusing to know what it means when you double and then bid again.

Here are the main 3 instances:

1) You make a takeout double and bid again
2) You make a negative double and bid again
3) You double their 1NT overcall and bid again

1) You make a takeout double and bid again

This shows the "big double" (approximately worth 17/18+).  Such an auction shows a "big" hand, but not too big, because partner can pass.

Typical examples of doubling their 1 opening would be:

 A2  
 AKJ98  
 AKQ  
 432.
 You double 1 and plan to bid hearts next.

 AQ10  
 K2  
 KQ76  
 AJ65.
 You double 1 and plan to bid notrump next (do not jump, though; just by doubling and bidding again, you show 18-20'ish).

 

2) You make a negative double and bid again.

This is somewhat the opposite of #1 above. Why? Because if you had a good hand you would just bid the first time (no negative double).

After partner's 1 and RHO's 1, you would simply bid 2 (forcing) with, say: 

 A2  
 AQJ876  
 AQ2  
 J2.
 

So, if you make a negative double and then bid a suit, it actually is a weak hand.  Say, partner opens 1 and RHO overcalls 1 and you hold:

 432  
 K108763  
 KJ32  
 --.
 

You make a negative double and partner (opener) disappoints you (as usual) and repeats his clubs (2). Now what? You can bid 2. How does partner know you are weak? Because if you had 10+ and hearts, you would have bid 2 the first time.

It is important to understand the difference between #1 and #2.

3) You double their 1NT overcall and bid again.

To understand this, you need to know what the double of a 1NT overcall means. It means "penalty/good hand." Also, if they overcall 1NT and you bid on the 2-level (for example, 

Opener	OVERCALLER	YOU (RESPONDER)
1	1NT	2

), the 2 bid is natural, NF (something like: 

 6543  
 KJ10873  
 2  
 Q2.)
 

So, if you double 1NT and then bid a suit, it shows a good (10+) hand. That is what you would have to do if partner opens 1 and they overcall 1NT and you have: 

 KQ2  
 KQJ987  
 J2  
 32.
 If you just bid 2 the first time, it would be weak and partner would almost always pass. You double first (penalty) and then bid to show this good hand.

SUMMARY: If you start with a double that guarantees a good hand and then bid again, you show strength. But, if you start with a negative double and then bid again, you show weakness.

If you would like to learn more about doubles, you might try Larry's Book on Doubles.

Or his Webinar on the Big Double.