The title "Urusei yatsura" is a pun within a pun. In normal Japanese, the title would be spelled "Urusai yatsura" (meaning "obnoxious people"). However, it 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 planet Uru" or as AnimEigo called it, "Those Obnoxious Aliens."
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...
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".
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.