SPECIAL in Fallout Shelter works slightly different than other Fallout games as Skills do not exist in this game. Each SPECIAL attribute increases dweller efficiency in various rooms, during combat or when performing certain tasks.
Strength is used to increase the speed in which power production rooms (Power generator, nuclear reactor) create power. It also determines how much a dweller can carry while exploring the wasteland. Dwellers can increase their strength in the weight room. It also increases the damage the dwellers can deal with weapons, at maximum level the dweller will always cause the maximum damage of the weapon. The more strength you have the greater chance of breaking into lockers you find in the wasteland.
Perception is used to increase the speed in which water production rooms (Water treatment, water purification) produce water. During quests it determines the speed of the critical hit indicator, higher perception decreases the speed the indicator moves, allowing you to get high critical hit damage more easily. In the wasteland it helps you to find locations.
These hit points are not retroactive.
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. To achieve maximum health, a dweller must equip a +7 uniform while having 10 endurance. At 11 or more, the dweller will be immune to radiation while in the wasteland (Note that this does *not* apply to real-time combat). Endurance can be increased in the fitness room. It is also your survivability in the quests or wasteland.
Dwellers start with 105 health points. They seem to get 2.5 + (endurance level * 0.5) health points each time they level up. So endurance level 17 will give you 11 extra points per level.
Charisma is used in radio studios and living quarters. Dwellers can increase their charisma in the bar room. Also used during encounters in the wasteland in order to help the Dweller succeed in encounters with "civilized" persons or groups, and allows dwellers to reproduce faster in the living quarters.
Intelligence is used by medbays and science labs. It can be trained using the classroom. In the Wasteland, the higher the Intelligence is, the better chance your Dweller has to succeed in encounters with wounded civilized persons (such as injured sheriffs, wounded slaves, etc.).
Agility is used in food production rooms (Diner and Garden) to produce food faster. It also determines how fast dwellers fire their weapons. Agility can be increased using the athletics room. In quests or in the wasteland it determines how many shots you can fire before the enemy fires or how much damage the Dweller can avoid when fleeing from lost battles.
Luck determines if a production room also yields caps along with the normal room reward (or extra caps during rush success). Luck also affects the chance of success of rush attempts, as well as chances of finding items when you go out on quests and explore locations in the wasteland. The higher the luck, the better the items. During quest combat it increases the speed the critical hit meter fills.