Ziggy and Roy Aladdin (portrayed by Gordon and Skip) were brothers that had an act that toured American and then the world.
When the Aladdin Brothers were lost in a desert in the middle east, a man on camel comes around and saves them. He magic reveals the entrance to a cave and asks them to go in and retrieve the magic lamp and bring it back to him.
When Ziggy retrieves the lamp, the cave starts to collapse on him. When Ziggy reaches the exit, Roy tries to negotiate for more compensation and angers the old man so that he causes the cave to collapse and trap both of them inside.
When Ziggy rubs the lamp, a genie (a la Johnny Carson) appears and says that he will fulfill their every wish. When they ask for a good place for their act, he takes them to Sheboygan. When the Princess arrives, everyone hides, as it is punishable by death to look upon the Princess. When Ziggy peaks and catches her gaze, Ziggy and Roy are apprehended by guards.
Ziggy and Roy trick the guards into getting their lamp back and escape imprisonment. Roy sends Ziggy off with the lamp and goes back to try to go before the Sultan and ask to marry his daughter. The Royal Magician (who was the old man that saved the Aladdin Brothers) suggests that he could be allowed to marry her if he presents the magic lamp as a wedding gift. When Roy arrives with the lamp, the Royal Magician steals it and attempts to take over Sheboygan, however, it turns out that Roy gave him a false lamp. The Royal Magician is taken away and the Aladdin Brothers become the new royal magicians.
In a twist, Roy ends up marrying the Princess instead of Ziggy. Therefore, Gordon steals Skip's girlfriend Clarissa by giving her a bigger part in the episode.
"Aladdin" is the main character in a middle-eastern folk tale, which is the most popular tale recounted in the anthology The Book of One Thousand and One Nights, also commonly referred to as Arabian Nights.