When the opponents make a takeout double of your partner’s opening bid they are asking their partner to bid their best suit.

This doesn’t really change much about how you respond to your partner’s bid except that it gives you the opportunity to show an invitational (or better) hand without raising the level of the contract, by redoubling.

As you can see from the panel above, weak hand with fit will still just raise partner’s suit by one level. But if you have fit and a better hand, you can show this by starting with redouble and then raising partner’s suit to the appropriate level.

Without fit, you can just bid in the same way you would have done without the double, but again, you have the extra option of  redouble to show some strength (about 10+ HCP).  Of course, since you can redouble either with or without fit, your partner will have to wait for your next bid to find out which it was.