To avoid spoilers, do not read
The Princess is who Tim is rescuing in Braid. Although she only appears in the final level, she is often referred to during the story and plays a major role.
The Princess seems to have been Tim's Fiance, or his friend, who has been taken away by the "Horrible, Evil monster". This is described in the books of world 2 describing them lounging in the garden giving names to the birds, shedding a small ray of light on the idea that they spent a long, happy time together before she was taken, thus beginning Tim's quest.
The Princess is a fairly thin lady of undetermined age, with long brown hair. She has dark blue eyes and wears pink lipstick. On her head she has a tiara (possibly made of some metal) and wears a long white dress and dark pink/purple shoes.
Much of the story centers around the princess and the sense of time. But in the end of the game, it is revealed that she is not to be rescued by Tim since Tim is the monster she is running away from. This happens when the player reaches the princess's house and is locked away, which happens if the player does not find all the hidden stars. But when the hidden stars are found (it is possible to reach candelabra and make princess explode without the stars, but it is quite a difficult task), the player can reverse time since two levers in the last level are glowing green, thus keeping them active after reversing the time so the player can use the candelabra to reach the princess. Reaching the princess makes her explode.
Possible Story reasoning
Due to the fact that world 1's time is reversed and by rewinding time in the "Untrue" ending, you can witness what really happened, a possible reason for the princess running away from Tim is because she is afraid that if he reaches her, she will die (from the explosion) and runs as far away as she can before she is saved by the knight. The "Horrible, Evil monster" she was snatched away from might be the "Boss" seen in worlds 3 and 5.