Maison Ikkoku - A manga turned anime romance/comedy by Rumiko Takahashi (creator of Ranma 1/2, Inu-Yasha, and the Rumik World series of shorts).
The manga was published in the early 1980's, and the TV show began before the manga finished it's run (though both storylines follow each other almost exact). In the U.S., the manga (published by Viz) just finished, while the videos are currently in flux (Viz is a little busy with Pokemon, I assume). It was, of course, fansubbed, so many of the English-speaking otaku have already had the chance to see it.
(Quick note: Maison Ikkoku is not shoujo. It's seinen romance, directed towards college-age boys, not young girls. It went a bit over the shoujo audience's collective head, though the shounen crowd seemed to like it and some seinen worship it like a religion.)
Takahashi generally creates stories full of fantasy and weirdness (aliens, gender-flipping, etc), though with Ikkoku she went for a more reality-based story: Godai Yuusaku is a 19 year old student (ronin) living in a boarding house (Ikkoku-kan) with several other zany, often drunk, comic relief tenants. At the start of the story (est. 1983), Ikkoku-kan hires a new manager (kanrinrin), the recently-widowed at 23 Otonashi Kyoko. Godai immediately falls for her, and the story takes off from there.
The plotline is actually fairly typical, and on first reading/watching it, it's easy to guess what happens next. But the heart of the show, what kept it going for so long, was the cast. The characters were so well developed and balanced that even though you knew what would happen, you still wanted to check up and see how everyone else would handle the twists. Every character down to the most minor role was so believable they could be neighbors (and after 96 episodes, one might start to wonder).
Another key to the success of MI was Takahashi's use of puns. Though totally obscure to anyone who doesn't know much of Japanese culture (myself included), the symbolism and jokes were a big draw to the original audience. For example, there are numerology references all over the place - all the characters in Ikkoku-kan had names relative to their room numbers (Godai - "Go" is 5, room #5). Another example is Kyoko's name - Otonashi means "gentle", but twist it a bit ("otto nashi"), it means "no husband." And there are references to Tokyo train stations and other local geography as well. There are documents all over the web researching Takahashi's cultural puns and symbolism.
Puns and symbols aside, the original or subtitled versions of Maison Ikkoku are also well-known in the anime community for their near-perfect use of language, making it a great help to people trying to learn spoken Japanese.
Opinion-ey part: When I first learned of this show, I was 19 and living in a looney apartment complex. It fit perfectly (except my landlord was this scary old guy, not a cute girl), and I was totally sucked in from there. It still holds my attention, and it's definitely my favorite story. For anyone who thinks anime is porn cartoons for dirty chronic masturbators, I can't recommend this enough to change your mind.
Related: Marmalade Boy, Kodomo no Omocha, Urusei Yatsura/Lum, Ranma 1/2
See also: Japanese Puns That Are Not Funny But At Least Are Puns