Boosters are unlocked on level 6. You can get Boosters from gifts or get them instantly with Diamonds. 
To activate a booster:

1. Tap on the booster tab on the right of the screen (the one with the booster icon, above the Gift tab)
2. A pop-up window will come up and in it you will see a list boosters (with the indication of the amount of each booster you have stored)
3. Tap the “USE” button if a given booster is currently stored or the “GET” button if you don’t have the booster and want to get it with Diamonds

Once a booster is activated in the previous screen it starts to do it’s magic right away! The descriptions of each booster can be read in-game by holding on the “i” button next to each booster, but here is the full list:

Happy Hour!:Earn double coins for curing all patients! Once activated works for 3h.
3h of Double Coins from Doctors:
Earn double coins for curing in the Doctor Rooms! Once activated works for 3h.
3h of Double XP from Doctors:
Earn double XP for curing in the Doctor Rooms! Once activated works for 3h.
3h of Double Coins in Treatment Rooms:
Earn double coins for curing your patients in the Treatment Rooms! Once activated works for 3h.
3h of Double XP in Treatment Rooms:
Earn double XP for curing your patients in the Treatment Rooms! Once activated works for 3h.
Happy Minute!:
Produce any cures without using up ingredients! Make sure your queue slots are free... Once activated works for 60 seconds!
3h of Double Coins from all Patients:
3h of Double XP from all Patients:
Earn double XP for curing all patients! Once activated works for 3h.
Produce any cures without using up ingredients! Make sure your queue slots are free... Once activated works for 1 hour! NOTE: This rare booster can be only won by completing Daily Quests.

Premium Boost:

Enjoy 6h of 2x XP and 2x coins from all you patients! NOTE: This rare booster can be only won by completing Daily Quests.
You can see that a booster is active by the change of the booster tab to show the currently active booster (at this point you cannot activate another booster before the first one finishes).