The Flaming Homer, also known as the Flaming Moe, is a cocktail drink, accidentally invented by Homer, and stolen by Moe.
Homer mixed eight types of liquors together (including tequila, Crème de menthe, and some kind of schnapps), and then, while not paying attention, he mixed in some Krusty-Brand cough syrup (Krusty's Non Narkotik Kough Syrup).
Homer returned to the slideshow, and Patty unknowingly dropped her cigarette ashes in Homer's drink, setting it on fire. Homer blew out the fire and discovered that the drink was much tastier.
Homer explained the recipe to Moe one evening, when he came in for a drink but the bar was out of beer. Moe then quickly claimed the drink as his own, renaming it the "Flaming Moe," thus stealing the idea from Homer.
The drink became extremely popular with locals and visitors alike, attracting a ton of new business based on nothing but the drink's addictive taste. Moe renamed his tavern, which had become "the place to be in Springfield" to the point where he was able to get Aerosmith to come in and play at the bar.
The drink also drew the interest of a rival bar, which hired Professor Frink to determine its chemical makeup. Frink was able to isolate all of the ingredients except for the cough syrup. His analysis machine claimed that the last ingredient was "love," which prompted Frink to exclaim, "Who's been screwing with this thing?" For the moment, the cough syrup was left as the secret ingredient.
Homer, meanwhile, had gotten incredibly angry at Moe, and jealous of how everybody adored the drink that he, not Moe, had invented. Homer showed up at the bar, laughing like a maniac at Moe and everyone, how they were "choking down your Flaming Moes, all the time wondering 'How does he do it?'"
Homer then revealed the secret ingredient to be cough syrup, in full hearing of the public. Everyone else in town quickly started serving Flaming Moes (under a wide assortment of varying names), and business at Moe's Tavern went back down to normal. Later, with no hard feelings, Moe served Homer a Flaming Homer on the house.