This article contains one or more spoilers. This is a potentially bad thing. Be careful reading on if you are not well-versed in the continuum or not caught up!
Fruits Basket is a manga that was originally created by Natsuki Takaya, and the later anime adaptation was directed by Akitaro Daichi. Both revolve around the Sohma family and an outsider named Tohru Honda. Over the course of the series, Tohru learns that the family has been possessed by the thirteen animals of the zodiac (the Cat, being the thirteenth animal, was tricked by the Rat into missing the induction feast, and thus is not recognized as an official member of the zodiac, though it still gets to possess people). Because of their possession, each time one of the members of the family is hugged by a person of the opposite gender, he or she changes into their respective animal. In the anime version, the comedy of this is played up a great deal.
Tohru is the main character of the series. She is a recently orphaned high school girl, who was living with her grandparents until renovations on their house caused her to have to leave. Instead of staying with her friends, which she feels would place too much of a burden on them, Tohru chooses to camp out in the forest close to the home of the Sohmas, though she doesn't know that last when she sets up camp. She is discovered by Yuki Sohma, a popular boy at her high school, and she offers to cook and clean for him and his cousin Shigure, who owns the house. Tohru is bubbly, happy, frighteningly optimistic, and entirely selfless. She is disliked by many fanwriters for precisely these qualities. She and Kyo end up in a relationship, and the final volume shows them old and happily married.
Yuki is the Rat of the Chinese zodiac. He goes to the same school as Tohru, and is known as "the Prince" to most of the girls. A group of them have even formed a fan/stalkers' club for him, though he obviously dislikes the attention. Due to his curse and the necessity of not letting anyone find out about it, Yuki spends much of his time at school trying to avoid letting himself be hugged by these zealous fangirls. Due to a traumatic childhood, Yuki has low self-esteem and feels isolated from most people around him. At first, he thinks he is in love with Tohru, but later realizes that he mostly sees her as a kind of mother figure. He eventually begins a relationship with another girl at the high school.
Kyo is the Cat of the Chinese zodiac. As such, he hates Yuki because it was the Rat which prevented the original Cat from being a real member of the zodiac. For all that, the two live together, though they do fight a great deal. Kyo is quick to anger and usually reacts violently when angry. He is initially wary of people, but gets over it quickly and actually makes friends with relative ease. Yuki states at one point that he is jealous of this ability, though it takes quite a while for him to actually tell Kyo so. Kyo is ashamed of his true zodiac form, which is a hideous monster, and runs from Tohru's proclamation of love because he thinks she will reject him. He is eventually talked around, and Tohru sees him in his zodiac form and does not reject him. There is great rejoicing. The end of the series shows Kyo and Tohru with their grandchildren.
For more information, along with minor characters, please consult the actual Wikipedia article.
Fruits Basket has a large following, and thus a large amount of badfic has been written. Many badfic authors dislike Tohru intensely and thus either insert Mary Sues in her place or ignore her completely and write slash. Kyo/Yuki is a common pairing, but any combination of the Sohmas has been written (see Rule 34). An agent working in this continuum should remember the curse (unless the fic takes place after the end of series, at which point the curse has been lifted) and should probably try not to activate it, as small animals are likely to escape and large ones, especially ones like the Ox or the Dragon, can be dangerous.