SPECIAL in Fallout shelter works slightly differently to other fallout games as skill do not exist in this game. Each skill increases dweller efficiently in various rooms, during combat or when performing certain tasks.
Strength
Strength is used to increase the speed in which power production rooms (Power generator, nuclear reactor) create power. Dwellers can increase their strength in the weight room.
Perception
Perception is used to increase the speed in which water production rooms (Water treatment, water purification) and determines the speed of the critical hit indicator, higher perception degrees the speed the indicator moves, allowing you to get high critical hit damage more easily.
Endurance
Endurance is used to increase the speed in which the nuka cola production plant creates food and water. It is also the assigned skill for the storage areas and determines how many hit points dwellers get as they level up. Endurance can be increased in the fitness room.
Charisma
Charisma is used in radio studios and living quarters. Dwellers can increase their charisma in the bar room.
Intelligence
Agility
Agility is used by food production rooms (Diner and Garden). It also determines how fast dweller fire their weapons. Agility can be increase using the athletics room
Luck
Luck is not used by any rooms but does determine if dwellers make extra caps during rush attempts, the success chance of rush attempts, as well as chances of finding certain items when exploring outside the vault.