The seventh ability score. Or maybe the eighth if anyone uses comeliness, or any other oddity out there. Luck is determined at character creation (or whenever the new ability score is implemented) by rolling 3d6 and adding them together. No modifiers, this is a straight up or down roll on the bell curve of fate. This number may never be permanently changed, even by a wish spell, but it can easily be temporarily modififed, eg a bless spell increases Luck by 1 while it lasts.
Luck is used to to determine the random chaos of the universe that can be adverse or serendipitous. When the unknown is about to happen, and there s no real way to gauge how it will go, bad or good, up or down, and it is specific to a characters destiny, then it may call for a luck check. “I leap into the darkness!” might be a good time to call for a luck check. Or maybe “I want to swing from the chandelier and land on his shoulders” along with a Dex check or two. The idea is to call for luck when the destiny of the character is involved. It could be something as small as picking out which chicken to buy from the market, or to choose between one door, hallway or another. Only when something hangs in the balance is a luck check needed.
To make a luck check, the character (or alternatively the secretive dm) then rolls a D20 and compares the roll to the characters Luck score. If the die roll equalls the (modified if appropriate) Luck score, or is lower, than the roll is a success. If the die roll is higher than the characters Luck, then they fail, ad something unlucky befalls them.
As always, when making an ability check, a NATURAL 1 is a CRITICAL SUCCESS and inspires some sort of awesomeness in proportion to the roll’s trigger. Naturally a 20 is a CRITICAL FAIL and should bring immediate doom to the character, or humiliation if only a minor infraction.
All these rules are meant to best represent the vagaries of luck and random chance in our mortal lives, and every rule has been carefully calibrated to bring verisimilitude to the continuously evolving game of dungeons and dragons.