Let me preface this by saying that Urusei Yatsura is one of my favorite anime of all time. The long running series was immensely popular in Japan and brought anime to new audiences. Urusei was one of the first series to use pop music, as well as references to both Western and Japanese pop culture. Lum, the female protagonist, became one of the first animated idols and inspired a whole genre of alien girlfriend/harem anime. Urusei was well received critically and won two Animage Grand Prix awards. Manga author Rumiko Takahashi is a household name in Japan and is responsible for other monstrously popular manga such as Maison Ikkoku, Ranma 1/2, and Inuyasha. Despite all this Urusei Yatsura remains relatively unknown in the west. If you consider yourself an otaku at all, I urge you to watch this show.
When aliens invade earth ordinary high schooler Ataru Moroboshi is chosen to challenge the invader's daughter, Lum, in a one on one challenge. Victory would have been assured for the Oni, but for one thing; Moroboshi's unfathomable lecherousness. In his moment of victory Ataru accidently proposes to Lum and the two are engaged. Now Lum's electric shocks are the only thing stopping Ataru from resuming his carefree life of girl hunting.
The series is highly episodic, and reminds me very much of The Simpsons in content and format. Every episode of Urusei Yatsura puts Moroboshi and his unfortuneate friends and family through some usually horrible encounter with aliens, spirits, ninja, etc. In it's 195 TV episodes, 12 OVA's, 6 movies, Urusei has pretty much done it all. The series influence can be seen on just about everything produced in Japan during 80's, and still influences things today. If you ever wondered where many of those Japanese cliches came from, look no further. There are a couple of slow episodes in the first season, but things only get crazier as more characters are introduced.
Urusei Yatsura really raised the bar in 80's animation, especially as quality steadily improves after the first season. Despite the occasional recycled background, Urusei consistently puts together beautiful characters, vivid and imaginitive sets, and frantic but believable action. The level of detail packed into every frame is matched by few series even today. The title and credits are mesmerizing, despite normally skipping them after watching a couple of times, I was unable to do so with Urusei. New titles are animated for each season. Of course if you don't like hand drawn 80's animation, this probably isn't going to appeal to you.
Urusei was perhaps the first anime to use pop music in its opening title and credits. The songs used, while having a blatantly 80's sound, are also undeniably catchy. Second season opener "Kokoro Bosoina" is an infectious piece of J-reggae that you can't help but nod your head to. The score is as quirky and interesting as the rest of the show. It never fails to set the tone for the hi-jinx that are sure to follow.
Ataru Moroboshi is indeed the most lecherous high schooler in the universe. His endless lust frequently drives the plot, but there are also dozens of supporting characters who mix up the action and keep things interesting. Many of the reappearing characters are so well developed and so interesting that they have become anime archetypes in their own right. From Cherry, the diminutive Buddhist monk with the bottomless stomach to the katana toting rich kid, Mendou Shutaro, you will undoubtedly recognize similarities to characters from more modern series.
Urusei is is definitely not your typical anime. While you should have some idea if you're going to like it or not after the first episode, the series really doesn't mature until the full cast is assembled. The animation quality also steadily improves throughout the series, so don't judge it on the first few episodes. Be careful though, or you may find yourself addicted and having to watch the entire series!