"Narrative Gaming" tends to refer to games that either give the players direct agency/control over narrative aspects of the game through mechanics, or whose mechanics are more directly tied to narrative conceits.
A powerful example can be found in various Powered by the Apocalypse games, which often give players who pass checks the ability to literally dictate the narrative/ensuing action: a character passing a knowledge check might result in the PLAYER explaining what their character remembered, altering the game world to match the new circumstances. (To put it in D&D terms, a Rogue could pass a Perception check, and 'learn' that the enemy has a weak leg, allowing them to inflict a slow with their next attack.)
FATE is another example, where the Aspect mechanic allows players to add details to scenes, and use them.
4e is a relatively low-narrative game, with a few notable exceptions. Player narrative control is the justification for Martial Dailies, for example: that a martial daily represents a sort of perfect confluence of physical energy, timing, and luck that the PLAYER can summon when they choose, rather than tying it to some complicated algorithm/roll chance.