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.
More than 1000 years ago she was faced with the tough task of having to exile her sister Luna after she had fallen to darkness because she felt like no one loved her. Because of this Celestia is often portrayed in the Brony fandom as being a troll who sends people to the Moon for fun. This is known as Trollestia
Thanks to a popular fanfic, Celestia is also seen as a molester known as Molestia. Because she had to exile her sister to the moon and rule alone for a 1000 years, she is automatically deemed a Tyrant, even when no one can prove so.
She has her own faction of the Brony fandom called the Solar Empire and along with her sisters fan faction the New Lunar Republic, continue to be a strong debate among bronies.