Ability Scores are one of the key mechanics near-universal to roleplaying games and war-games. Defining various statistics or proficiencies of a character, ability scores provide a testable medium to compare against - basically, ability scores are what makes something /tg/ worthy as opposed to just a game of "let's pretend".
Dungeons & Dragons has one of the most iconic line-up of ability scores, in the form of Strength, Dexterity, Constitution, Intelligence, Wisdom and Charisma. This line-up is so iconic that most other RPG ability score sets boil down to renaming and sometimes expanding (or, more rarely, contracting) this list.
In war games, ability scores are used to keep combat flowing smoothly. For example, in Warhammer and Warhammer 40000, you have the ability scores of Movement (how far a unit can move in a turn), Weapon Skill (likelihood of hitting with a melee attack), Ballistic Skill (likelihood of hitting with a ranged attack), Strength (ability to inflict damage), Toughness (ability to withstand damage), Wounds (how much damage it can take), Initiative (how fast its attacks are) and Leadership (morale).
See also stats.