~STORY~
(Contains spoilers, so don't read if you haven't read the manga / watched whole the TV series yet. Or do read, but you've been warned)
To begin with, Urd is one of three Norns and her dominion is Past. But in the context of AMG universe it's almost nothing but a nice title. In fact, Urd is the System Administrator of Yggdrasil, the OS of the Heaven. Together with her youngest sister Skuld she does major maintenance works. Well, when they're both not on Earth.
But actually Skuld and Belldandy are only half-sisters with Urd, as they have the same father. There is no information yet of who he is, and the most popular theories are that he is either some very high-ranking god or Kami-sama (Almighty) himself. The second theory comes from the Dark Horse translation, where goddesses call the Almighty "father", but in the original version they always call him Lord or Kami-sama.
However, the theory about the Almightly being Norns' father has some support, since we know that Urd's mother, Hild, is in fact the Damakaicho, the Great Ruler of the Demon Realm, or, to say it shorter, the Queen of Hell. We don't know why, but she sent her daughter to Heaven at a very young age (she was about 5 or 6, according to the TV series, episode 21) and it seems that they haven't been in contact ever since (until the recent events in the manga). As I said, her motives are unknown, and I personally think that she just had no choice, but for Urd it was a shock.
So she got to Heaven and met her younger sister Bell. Several years later Skuld was born and three of them grew up together.
As a teenager, Urd tried to get a First Class license, but the was advised by the examiner that it woud be best if she didn't become a First Class goddess. Others though it was because of her half-demon blood, but Urd herself knew the true reason. However, she never admits it in public, and always says that she failed the exam because "liars are inelligible".
When her younger sister Belldandy went down to the Earth, making a contract with a mortal guy Keiichi Morisato, Urd decided to keep an eye on them. To her disappointment, after a year of living together they haven't even normally kissed each other, so the big sis decided to help them with their relationship. She came to their house and started flirting with Keiichi "just to teach him techniques that will let him do this and that to girls". But her pushy attempts only lead to a big arguement with Belldandy. Their conflict was, however, soon interrupted by a lightning bolt which struck Urd and left a message from Heaven, which stated:"It has been determined that a recent temporary system crash was due to interference by goddess Second Class (Limited) Urd. Therefore she must remain in exile until further notice". So she had no choice but to stay on Earth.
Currently she's living with her sisters, Keiichi and the demon-cat Welsper in the Morisato Residence (which is actually an old buddhist temple). She mostly spends her days watching TV or working on a wide variety of (usually love-related) potions in her chemical lab.