Even though Ran is only a boss in two games and has never been a playable character, she's a fairly popular character and is often seen in fan comics partly because of her master Yukari, who is another popular character.
Ran is supposed to be a 9-tailed fox (called a "Kyuubi") which, according to ancient Japanese folklore, would be at least 800 years old. Foxes start life with 1 tail and gain an extra tail once every hundred years. So at 100 years old they would have 2, and at 800 would have 9. At this point, assuming Ran follows traditional folklore, she will eventually start losing tails until she has 4 and will be considered a "Tenko" (heaven) type of fox and will then be one step away from divinity.
Ran is very good at math, being able to solve complex equations in her head.
Ran is the only stage mid-boss in any Touhou game that is immune to bombs.