A long-cage (Mtn bike) RD (rear-derailleur - please learn to use this abbreviation. Our carpal-tunnel syndrome thanks you!) will help you shift a larger range of gears. This allows you to use all your gears with better smooth shifting, rather than having your shifts become spotty and difficult. Stressing your drive-train and causing you problems. The Deore LX/XT mentioned is a good, mid-cost, choice.
The Shimano Tourney is a low-end RD. You can do better - such as the Deore offerings mentioned. You could also buy one that's twice the price for proving to people that you're rich and sexy... count your silverware when she/he leaves.