Personally, I recommend the MDA (Mechanics, Dynamics, Aesthetics) approach and looking at the 8 Aesthetics( Main Reasons why people play games):
Game as sense-pleasure (Hearing/music - Catherine, Sight/art - El Shaddai )
as make-believe (Soldier - Call of Duty, Spy - Splinter Cell, Mage/Warrior/Rogue - Skyrim, etc, etc)
Game as drama (For the story, Most J-RPG's, Spec Ops: The Line, etc)
Game as obstacle course (Super Meat Boy, Demons Souls, Any game that's hard)
Game as social framework (Teamwork with other people. Horde on Gears of War, Co-op mode on any game, etc)
Game as uncharted territory (Wandering around Skyrim, Fallout 3, Sandbox games)
Game as self-discovery (Allow's a person to express themselves. Through choices, in-game creation - minecraft, music creation in Mabinogi, etc)
Game as pastime (To lose oneself in, mindless past-time. Pokemon, Final Fantasy, etc)
Personally, I have one more:
Game as competition (Multiplayer in Call of Duty)
These are the reasons people play games.
Most games have at least three aesthetics. Although some games have fragments of other aesthetics, Mario having story to rescue princess, nobody plays Mario for the story, thus Narrative isn't an aesthetic in Mario.