Ataru Moroboshi's name literally means "to be struck by a falling star" i.e. someone so unlucky that if there's a one-in-a-million chance of having a calamity strike, he'll be the one to get it. This is alluded to in the second "eyecatch" commercial bumper used, where Ataru is hit ont he head by the "hoshi" kanji in the "Urusei Yatsura" name (see other trivia for this)
The name "Urusei Yatsura" is a pun within a pun. In normal Japanese, the name would be spelled "urusai yatsura" (meaning "obnoxious people"). However, the title is spelled with the Japanese kanji karacter for "star" ("Hoshi" or "sei") making it "Urusei Yatsura" which could be interpreted as "jerks from Planet Uru," "Those annoying bastards from the planet Uru" or as Animeigo had it for a while, "Those Obnoxious Aliens."
When spoken out loud, Cherry's real name, Sakuranbo, sounds just like the Japanese word for "cherry", or the fruit. However, when reading the kanji (Japanese characters) used to spell his name, the meaning can be interpreted as "Demented Monk"; a very fitting description, indeed.
Was, for its time, one of the most popular anime shows in Japan. It even succeeded in forcing "Captain Harlock: Endless Oddessy SSX" (the spin off to "Arcadia of My Youth") off of the air for lack of viewers in its timeslot!
The "Dappya Monsters" who appeared in many episodes were not actually intended to be cast members of Urusei Yatsura, but instead are a complex in-joke started by Takahashi in the manga and carried over into the TV series. They were the alien invaders in Takahashi's first work, "Kattena Yatsura!" ("Those Selfish Bastards!") a manga story that got Takahashi her contract with Shogakukan, where she started work on Urusei Yatsura.
Megane, Perm, Chibi and Kakugari (Lum's Stormtroopers) originally appeared in the manga for a very short time (and were never directly named), after which Takahashi thought that the gag had run its course and quietly retired them in favor of more dimensional friends for Ataru. The writers of the TV show took the characters and stretched their obsession to Lum to the nth degree, making them into parodies of obsessive fanboys all over the world. Their names, by the way, mean "Four-eyes (lit. "eyeglasses")", "Permanent wave (his hairstyle)", "Runt" and "Crew Cut" respectively.
Lum's manner of speaking ("uchi" meaning "I" and the ending "datcha") was copied by Rumiko Takahashi from the speech patterns of Japanese high school girls of the period. To a Japanese ear, Lum sounds a little like a "Valley girl".
The oni reference Japanese mythology. In many Japanese tales (including "Momotaro, the Peach Boy") Oni were analogous to demons, and were known for committing horrific acts on humans. They were described as having a horn or horns, being primitive and wearing animal skins. In some myths, they also had control of the weather. Male oni were hideous and violent, where as oni women were beautiful but posessed of a fierce temper and jealousy.
Ataru's task to grab Lum by the horns in the first episode comes from a folk belief that if you can catch an oni by the horns, he (or she) is obligated to grant you a wish. Therefore by Ataru shouting "I can get married" while holding Lum's horns, he is inadvertently making a wish. And of course, Lum misinterprets just whom Ataru wants to marry...
The story of Yoshitsune ("Ah, Father, you were Strong") is a direct parody of the mythologizing surrounding Minamoto Yoshitsune, the younger brother of Minamoto Yoritomo, the first Shogun of Japan beginning circa 1185. Yoshitsune later was hounded to his death by Yoritomo, although legends persist that he actually fled Japan for Mongolia and became none other than Genghis Khan.
Lum's ex-boyfriend Rei transforms into a giant tiger-striped cow who strikingly resembles the Toho kaiju Baragon from Frankenstein Conquers the World, Destroy All Monsters, and Godzilla, Mothra, and King Ghidorah: Giant Monsters All-Out Attack.