Redoubles

These can have 2 distinct meanings.

1. 1♥ (us) * (Them) ** (us)
This redouble shows:

9 or more (generally 10) points,
the ability to penalise the Opposition in at least one suit.
it will not generally have more than 2 in Partner's suit,

so will be something like: ♠ A 10 7 2 ♥ Q 2 ♦ A Q 10 2 ♣ 10 9 8.
If the Opposition bid spades, or Diamonds, you will Double again. - This is an optional double, showing at least 4 trumps in one of their suits, the balance of points, and is prepared to defend the contract. Be aware, if you have a fit with Partner, there is more likelihood that the Opposition can make their contract!
Holding : ♠ A 10 7 2 ♥ Q 8 2 ♦ A Q 10 2 ♣ 9 8 Redouble, by all means, but then bid 3♥, to show a 3 card ♥ support.

So the important factor, is that the person who Re-doubles hasn't got a more satisfactory bid e.g.:

Partner opens with a bid of 1♥, and your right hand opponent doubles:

You hold:

♠ K J 7 2 ♥ 10 5 ♦ A Q 7 5 ♣ J 5 3 You haven't got a natural bid (any suit shows at least 5 cards), so you could not:

bid 2NT (this should show a good raise to 3 hearts),
bid 3♦ (should show at least 5 diamonds, and deny heart support),
bid 1♠ (should show 5 spades, and show no trump support),
bid 2♦ (weaker than 3♦),
bid 1NT (you have too many points for a bid that only shows 10 maximum)

Thus ** (Redouble) is the only bid that correctly expresses the value of your hand! - i.e. no natural bid (5 card suit), no suitable 'No Trump' bid.

If you hold:

♠ K J 7 4 2	♥ 5	♦ Q 5 4 3	♣ A J 3	You should bid 2♠ - Natural, and forcing for one round.
♠ K Q 8 7 4 2	♥ 5	♦ 8 7 4	♣ J 4 3	You will just bid 1♠
♠ K J 7 2	♥ 10 8 2	♦ A Q 7 4 2	♣ A	You will bid 3♦ - again forcing for at least 1 round.
♠ Q 7	♥ 10 8 2	♦ 10 9 8 7	♣ K 8 7 2	You would pass
♠ 7	♥ J 10 8 2	♦ A Q 10 7 4	♣ K 8 2	Of course, here, you would raise immediately to 4♥.
♠ A 8 2	♥ K 7 6 5 2	♦ Q J 5 4 2	♣ ~	How about a splinter bid of 4♣. To be discussed !!!!
♠ Q J 6	♥ Q 7	♦ K 8 7 2	♣ Q 10 7 4	Bid 1NT - showing a maximum of 9-10 points.

2. However:

You	Opposition	Partner	Opposition
1NT	*	2♣	Pass
Pass	*	**	Pass
???

??? What this is showing, is a hand which is trying to rescue into a better contract :
♠ 10 8 7 6 ♥ Q 10 9 2 ♦ J 4 3 2 ♣ 4

2♣ is a sort of Psychic bid, - you are happy to play with a singleton club opposite Partner, so long as you are not Doubled (reasoning, the Opposition have 23-25 points, and a likely game); once the Opposition have doubled 2♣, you redouble. This is for take-out. If you were happy to play in 2♣* you would Pass!!