This is an older series, but even so its status as one of the foundations of anime is probably well assured. A series does not run for 200 episodes without considerable public support. It also marks the establishment of Rumiko Takahashi as a well-known name.
However the series is one that many have tried and found themselves confused or apathetic towards its supposed charms. Partly this is the age of the material, partly some wobbly production in the early seasons but also because of the nature of the material. It's very Takahashi mind you, but less polished and immediately accessible than her later works.
The essence of Urusei Yatsura is domestic Japanese life (in my opinion). We get to see families, mundane aspects of Japanese life and lots of normal activities such as school, lessons and family dinners. In Urusei Yatsura parents and grandparents may have suprisingly active roles in the story. Now add in Takahashi's exceptional abilities to craft characters. They're all a little bit weird, the women even more so than the men, but they also have charm, depth and may intrigue you as to what makes them tick. And the cast is huge, not only the regular cast but the number of occasional and episodic characters. And they're all woven into a complex tapestry full of likes, dislikes, shared character elements or past hostilities. There's some lovely dialogue, scenes and even a touch or two of romance to provide a wonderfully tangled environment full of intriguing characters.
And then, to make it more fun, drop something very weird into the middle of it and watch the fireworks. In this case the element is a super-cute alien female who, through a misunderstanding, falls for and considers herself the wife of the male lead. For his part he's a lecherous, phenomenally unlucky guy with a very weird psychology. He's often called an idiot, but it's more complex than that...it's just that he perceives reality in a fairly strange way, and is totally focused on living according to this weird model. And the last thing he wants is a stable marriage to such an exotic girl. Although the fact that Lum zaps him (she can fly and generate lightning bolts naturally) whenever she get's a bit emotional (happy and angry) also has some small part to play. But what really scares, but also tempts, ataru is that she honestly does love him completely...something his strange outlook isn't really capable of dealing with.
The ensuing conflict of personalities is one thing, but with her ability to attract chaotic weirdness from space, and his matching ability to call forth every strange element on earth, there's no shortage of weirdness going on. And of course this reaches out to affect the rest of the cast. It is the mixture of insane situations, complex cast against the background of mundane Japan that makes this show so much fun. Takahashi's inventiveness, some elements of parody, plus bits of sci-fi and mythology keeping the pot bubbling. And I really must emphasise that the stories are quite willing to break reality, or introduce incredibly impossible elements, if there's some fun in it. Dimensional travel, Time travel, Fate, Magic, Alien invasion...it's all just another `normal' afternoon at Tomobiki high.
As a result trying to describe this series is largely doomed to failure. In 200 episodes there's a huge amount of material and characters covered. And the interactions of plot and character are quite complex. I will however make the point that it does take a while to get into. This is partly because originally Lum was going to be the `tormentor' of Ataru, although this (wisely) ended up more as a comedy/romance. Also the situations become more fun when theres a lot more characters involved and we get to know them better. Indeed both the characters and the environments seem to continuously grow in both number and depth. The stories themselves also tend towards light-hearted fun, this is definitely meant to be a comedy, and some may find this tiring in large doses.
I will also mention that the anime is considerably different from the manga. The manga is more subtle and restrained, whereas the anime tends to attempt to be a bit more manic and loud. It also loses some of the character elements in the translation. In addition the show does change over the durations of it's run. With the help of some Usenet responses i'll mention the following important ones. The first season (episodes 1-21, plus the spring break) is the least liked. It largely copies the stories and structures from the manga, but with less subtlety, and the stories don't always transfer well. They also tend to be short, often leading to two stories per episode. After this point the creators start to take the manga more as an inspiration only, and grow the characters along slightly different parts, which makes the episodes somewhat more satisfying. It also means they're far less true to the manga, but this seems an acceptable trade off in general. It was also mentioned that as the series progresses it began to gain more and more support in Japan, leading to bigger budgets for both story and animation. Thus many will agree that some patience is needed to really get into Urusei Yatsura.
For myself I find it occasionally disappointing, some of the humor and animation having aged badly. I'm also quite a fan of the manga, and find many of the changes introduced in the anime series to be against the spirit of the manga. It also means that the element of suprise, which is interesting considering the weirdness of the plots, is lost. On the other hand the characters are wonderful and their interaction and dialog excellent fun. They're so complex you always want to know more about them, and this only increases the more you learn. They're also strong, with many of these characters being sufficiently memorable they could easily star in their own series (but then i'm a shinobu fan). In addition the insight into Japanese customs, domestic life, and mythology is both exotic and intriguing to me. In short it's like much of Takahashi's material, it often seems a bit light and directionless, but it's also terribly addictive and capable of moments of great beauty. I'll stick with it as long as I can find tapes to watch. Mind you, my preferred form will remain the manga.
The animation is fairly dated. It also copies the appearance of the manga, and some may find Takahashi's style a bit unusual, and perhaps too cartoony. It also often wants to represent complex scenes, or quite cinematic scenes, but doesn't have the budget to do so. This is probably worthy of respect, that it's willing to be so ambitious with what it aims for, but it's not always successful. That said you do adapt to the style and there is a lot of character and humor in the character animation. It, generally successfully, merges some hyper-energetic action sequences with some beautifully subtle expressions and dialogs which is quite impressive. There's also lots of hidden jokes, puns (which we lose), cultural references (which are enjoyably confusing) and visual humor. It also has excellent voice talent, which is important in giving the many characters such a strong sense of personality. It also has a lot of pretty enjoyable pop music and some dated but fun BGM.
If you want to know more, about the immense amount of material i've skipped over or not mentioned at all, then try Tomobiki-cho which is a wonderfully detailed and authoritative web-site on all things Urusei-Yatsura related.