Princess Celestia is one of the two rulers (her co-ruler being her sister Princess Luna) who preside over Equestria, the fictional land in which the My Little Pony show is based. Princess Celestia functions as the head of state during daylight hours, and has many responsibilities including raising and lowering the sun each day. She also acts as a mentor / support figure to Twilight Sparkle and friends.
There has been no mention within the show as to why she is a princess and not a queen, but it has been unofficially acknowledged that this comes across as more 'friendly' to the target audience of the show.
Despite being posessed of formidable magical abilities she does have her limits as evidenced by certain episodes of the show. As with her sister she is also attributed with immortality.
She has a kind disposition and many positive attributes; but sadly she lacks a lot of character background, so more detail is difficult to provide.
Princess Celestia is remarkable for being one of only three ponies with both wings and a horn. Her stature and status guarantee she stands out wherever she goes.
Princess Celestia is a pony well versed in the magic of friendship.