Reimu Hakurei is one of the main characters of the Touhou Project series, having appeared in almost every game in the series. As the sole maiden of the Hakurei Shrine, she is commonly called upon (or calls it upon herself) to investigate strange occurrences throughout the world.
In earlier games, Reimu's spiritual abilities were highly limited. (In Highly Responsive to Prayers, she was in fact unable to harm enemies directly at all.) As time went by, however, she developed her abilities, becoming able to use the Yin-Yang Orb, special amulets, and her own (limited) spiritual power in order to fight more effectively. Sometime between Mystic Square and Embodiment of Scarlet Devil, this culminated in her developing the ability to fly under her own power. (Before then, she relied on Genjii, her turtle familiar, to fly, and in fact did not fly at all in HRtP.)
As her abilities have changed, so has Reimu's personality. She is commonly described as easygoing, curious, at least somewhat dutiful (at least, the shrine hasn't fallen apart due to negligence on her part), and optimistic —sometimes overly so. Her personality seems sympathetic toward many individuals; there are few people indeed who remain her rivals past the first game in which they appear (though this is not to say she doesn't get into fights after the first meeting), and the shrine has become a moderately popular hangout for some of Gensokyo's more interesting female individuals.
In Undefined Fantastic Object, she seems very intent on getting the treasure ship's riches, probably due to the events following the Hakurei Shrine's repair and re-opening in Scarlet Weather Rhapsody.
In Ten Desires, a lot of divine spirits start to congregate and due the fact that is an "Incident", Reimu has to go and figure out what drove them to spread.