Tenchi Muyo, an original anime series created by Pioneer in the mid '90s, is one of the original fan-favorite megaseries. Its wacky stories, hilarious situations, lots of violent-but-pretty girls, and a slightly off-kilter sense of humor endeared fans to the series from the first episode of the OAV series that started the whole mess. And it is a mess, consisting of about five entirely different continuities involving the same basic set of characters; we break it down here to relieve some of the confusion.
The title, incidentally, is a multi-layered pun, but in Japanese is mainly recognizable as the common label "This End Up."
Tenchi Muyo is the ongoing story of Tenchi Masaki. Tenchi is one of those anime guys that has more luck with the ladies than anybody deserves. Well, luck might not quite be the word... Tenchi's admirers aren't exactly the cute, garden-variety high school girls that you'd expect. Tenchi ends up living with his would-be wives, the Princess Ayeka and her arch rival the former space pirate Ryoko, along with a whole slew of other women (not to mention his father and grandfather). And, of course, these folks have a tendency to attract intergalactic villains. Tenchi's life will never be the same...
Of course, the details of the story and backstory depend quite a bit on which continuity you're looking at, but the basic setup is more or less the same across all of them except the Pretty Sammy spin-off.
The high school student at the center of way too much attention. Depending on which series you look at, he's either a relatively normal guy (by anime standards) stuck with some abnormal roommates, or an unknowing descendant of the Jurai royal family. He helps care for the shrine that his grandfather maintains, and has (until the series starts) lived a relatively quiet life in the country.
Crown princess of the Jurai royal family, destined to become the future ruler of an interplanetary empire. Ayeka got stuck on Earth when her ship crashed. She immediately fell in love with Tenchi, and has dedicated herself to making him return the affection... and getting Ryoko out of the way. She's pretty, refined, and has a vicious temper when it comes to Ryoko.
A thousand or so year old space pirate known for inflicting planetary-scale destruction in the process of raiding civilizations for their valuables. But Ryoko isn't so bad; she's got a hot body (and isn't shy about using it), and she's got a major soft spot for Tenchi. She's also got a faithful sidekick, Ryou-ohki, a cute, furry little half rabbit/half cat that just happens to be able to turn into a very large, very powerful spaceship... that meows. (Actually, it "miyaa"s, but that's getting technical). Anyway, Ryoko's stuck on Earth too, and is willing to do just about anything to catch Tenchi's heart (or, failing that, his body).
Ayeka's cute younger sister and permanent perch of Ryo-ohki; she's younger (or looks younger, anyway) than the rest of the group, and is the household cook.
A semi-mad scientist and the self-proclaimed greatest genius in known universe; she's thousands of years old but currently inhabits a rather diminutive body. She lives in her lab--a vast complex located in a dimensional pocket (the door's under the stairs of Tenchi's house).
Wayward Galaxy Police agents; Mihoshi has a nose for trouble and a brain for... not much. Kiyone is Mihoshi's considerably less ditzy partner. She was not a regular character in the OAV, but was mentioned as Mihoshi's former partner.
Finally, there are the two elder members of the Masaki clan, Tenchi's widowed father and (despite appearances) maternal grandfather; the latter cares for the shrine that Tenchi will probably inherit (if everyone else doesn't kill him first), and the former owns the house that this whole crew has taken up residence in.
Or, more accurately, complete lack of continuity. As mentioned, there are between three and five entirely separate continuities, depending on how you count.
The first would be the Ryou-ohki/OAV continuity; it encompasses the three Ryou-ohki OAV series (two older and one about a decade newer), the Mihoshi special (though that's only a loosely-related a side-story), and possibly the second movie (though it includes Kiyone and doesn't apparently fit). The comic series (or both comic series; the "new" one is just the next part) is also in the OAV continuity, and is intended as a direct continuation of the story. Finally so is the GXP TV series; although it follows a completely unrelated Tenchi-lookalike earth kid who joins the Galaxy Police, it does belong in the OAV continuity.
The first TV series is a direct retelling of the OAV series with slightly altered characters and, unlike the OAVs, a definite end. The first and third movies (maybe the second, too) appear to follow the TV continuity as well, though since they don't quite fit anywhere in the series you could also claim that they belong in a continuity of their own branching off the TV series setup.
Finally, there is the Tenchi in Tokyo TV series continuity. It seems to pick up in the middle of the series and assumes some familiarity with the characters, but actually follows it's own unique continuity, with totally different origins for all the characters explained in a later episode. This series chronicles Tenchi's adventures as a high school student in Tokyo with yet another pretty girl, Sakuya, after him (this one's not so violent, though).
Oh, and of course there is the Pretty Sammy spinoff series; they use the same basic set of characters without the crazy backstories, and change Sasami into a Sailor Moon-style magical girl heroine. There's also a new series, Sasami, attached to this spinoff franchise.
If you're wondering, it's best to start your Tenchi obsession--and once you start, obsession it will likely become--at the beginning of either the first TV series or the OAVs. The story and characters will make a lot more sense that way, and you won't spend the first few episodes (or the whole movie) trying to figure out who these freaks are. Wherever you choose to start, though, Tenchi Muyo is kinda like potato chips; it's hard to get enough.