The Horse
The seventh zodiac animal.
In Fruits Basket, certain members of the Sohma family are cursed. They are possessed by the spirits of the animals from the Chinese zodiac and when weak enough will transform into the animal they are possessed by. The curse also changes them into their animal forms if they're hugged by an uncursed member of the opposite sex.
In Rin's case, she is possessed by the spirit of the horse. This allows her to run at incredible speeds; she's so fast that no one else has any hope of catching up to her when she decides to run free. She would also get along very well with other horses and they would be drawn to her if any were nearby. Since she lives in such an urban environment, there's very little chance of that happening unless she was at a zoo!
Takaya Natsuki has stated that she always wanted to design Rin with short hair but felt she needed long hair to reflect her status as the horse.
IN JAPAN
In Japan, the animals of the zodiac are not only dedicated to one year out of a 12-year cycle but also reign over the different months in the year. Since the lunar calendar year begins with November, May ends up being the month of the horse as it's the seventh month in the cycle. (The horse in Japan is the seventh animal to arrive at the banquet.) Rin's name comes from "Isuzukurestuki" which is the sixth month of the year... I'm not sure why Takaya Natsuki didn't use the name for the seventh month instead ("Fumihirogetsuki"). Perhaps it's because she skipped Yuki, the only one of the juunishi not to have been named after a month, and went straight to Hatsuharu? Either way, I'm glad she used Isuzu; it's a very pretty name for Rin. :)
"Hinoeuma" is a special, ominous date in Japan. Every fifth year of the Horse is called Hinoeuma and it's believed that if a woman is born in this year, she will be temperamental and cause her husband so much stress that he'll die earlier than he would otherwise. The last Hinoeuma was in 1966 and in that year, far fewer babies were born. I don't think this applies to Rin at all but I thought it was an interesting fact about the year of the horse. :)
PERSONALITY
According to this site, it is said that someone who is born in the year of the horse may be stubborn and blunt but also very patient when listening to other people. They're also extremely loyal to those close to them and are capable of solving conflicts. Regarding love, they tend to be the passive one in the relationship, content to let their lover take the initiative.
Rin seems to fit these personality traits; she's incredibly stubborn in her drive to end the curse. She's also rather blunt and straight-forward with people. She tells Kagura to move on from Kyo, tells Hiro to stop worrying about her, etc. We also see how loyal she is to Haru and Tohru. Rin also seems to be the more passive on in her relationship with Haru, as it's Haru who initiates the relationship and pursues Rin.