Strength measures your character’s muscle and physical power. This ability is especially important for fighters, barbarians, paladins, rangers, and monks because it helps them prevail in combat. Strength also limits the amount of equipment your character can carry.
You apply your character’s Strength modifier to:
-Melee attack rolls.
-Damage rolls when using a melee weapon or a thrown weapon (including a sling). (Exceptions: Off-hand attacks receive only one-half the character’s Strength bonus, while two-handed attacks receive one and a half times the Strength bonus. A Strength penalty, but not a bonus, applies to attacks made with a bow that is not a composite bow.)
-Climb, Jump, and Swim checks. These are the skills that have Strength as their key ability.
-Strength checks (for breaking down doors and the like).
-Carrying Capacity
-Grapple and Bull Rush checks
Frontline melee characters will want a Strength score of around a 14 minimum, and this is usually recomended to be the highest stat.
Other characters usually put this stat at 14 for the highest, usually more towards 12 or lower. It is commonly the lowest stat for arcane casters and druids, neither of whom make much use of it at all.
For other uses of Strength, Check out this thread on the WoTC boards X Stat to Y Bonus