In the Manga, we are first introduced to Urd in Part 1 Volume 2, "Sexy Sister," when Urd sends a tape of herself to Keiichi in a porn video. Keiichi watches the tape, and Urd emerges through the television set, which, by the way, is her method of manifestation. In the Anime, she is first introduced in Volume 2, "Midsummer Night's Dream," where she also sends a tape of herself in a porn video.
In Part 1 Volume 2 of the Manga, Belldandy states that Urd is much more powerful than her, but that she's been so 'naughty' that her license has been limited. Of course, as of late, it seems Belldandy may be the strongest goddess (she is First Class, after all). It's hard to say who has more power (just because Urd can't always control her power, doesn't mean she is weaker), but we do know that Urd has poor precision, which is why she's a Second Class goddess.
Urd is a half-goddess, half-demon, obviously having the mother as the demon. This makes Belldandy and Skuld her half sisters and explains why Urd's skin is darker, and why her actions are often questionable. Belldandy states in both the Anime and Manga that goddesses can never tell a lie, so when Urd lies, Keiichi figures she is telling the truth because of what Belldandy says. Little did Keiichi know that Urd lies often, and has her license revoked (in the Manga) because of it. Despite her demon-like habits, she shuns the darker side of herself, and strives to be pure.
Urd is the series' "Cupid" or "Love Goddess." She is constantly pushing Keiichi to 'make a move' on Belldandy. Time and time again she uses potions and magic spells to enhance the speed of their relationship, but as always, something goes wrong and it backfires. Despite her screw-ups, Urd's sisters are always in her best interest.
Magic potions and herbs are her specialty, even if they don't always work the way she wants them to. Her room is filled with various concoctions that she organizes and sorts through on a daily basis. It seems to be the only thing that Urd is devoted to. She even has a count of every pill in every bottle, so she knows exactly when one is missing (Manga). For more information on Urd's magic, go here.
On a number of occasions, Urd's demon side emerges and takes control (Manga). Disaster always follows these occurances, as does Mara. Of course, even when Urd is in her full-demon state, you can always count on her whimsical nature to leave you with a good laugh.
When Urd is in her goddess form, she has two triangles underneath her eyes, and in the middle of her forehead. When she's a demon, the triangle on her forehead is replaced with thin diagonal strips (\/). These marking show a goddess' or demon's relationship to Yggdrasil and the Niflheim (the evil equivalent to Yggdrasil).
Urd is Yggdrasil's System Maintainence Officer. She helps Skuld debug and run the system. However, when she hears of Belldandy's ventures with Keiichi on Earth, she leaves and messes with their relationship which in turn messes with their contract. She is banished to Earth for her bad behavior until further notice.
Her main hobbies include making love potions (or any potion for that matter), messing with Skuld, helping others with their love life, watching t.v. (particularly soap operas), and drinking excessive amounts of sake, which happens to be her alternate source of energy. You can basically sum up from this that Urd is lazy, irresponsible, and cares nothing for manual labor. Any chore that she performs is done with her powers or some sort of potion. She also has a love for singing karaoke.
Urd's main weakness is falling asleep whenever she hears polka music (Manga) which is a method used by Mara on a number of occasions.
Urd mostly causes trouble for Keiichi, but Urd can be helpful, and has gotten them all out of sticky situations on a number of occasions. While Urd can't control her powers, and is always scheming something (along with Skuld), she always (at least most of the time) has good intentions.