There are so many theories flying around here that I just thought I'd throw mine in (even if the thread hasn't been posted in for about two months).
What I think happened is that once Azulon told Ozai that he would have to kill/exile his own son, Ozai refused, extremely angry at the prospect.* Ozai most likely thought the prospect ridiculous, he most likely wasn't planning on having more children, and with the Fire Nations tendency towards a male Hierarchy, this would have destroyed any chance he himself had at continuing the line if he was put into power. Most likely he had gone there with the original intention of persuading his father to name him as heir, but with the prospect of loosing his only son, he flew into a rage and attacked Fire Lord Azulon, killed his own father. And then, with the Fire Nation not being known for it's just system of government but known for it's thirst for power, Ozai bribed the first few people on the scene with positions of power, promising them that if they helped cover up what he had done, and vouch for him when he says that Azulon named him his successor.
Then, he went to his wife and happily told her that he had managed to convince his father to name him heir, but sadly, shortly after that he passed away. Ursa, being the smart woman that she is, most likely put two and two together and got that her husband had killed his father. She would have stayed, if not for the fact that she soon discovered something else: Ozai's plan for Zuko. Ozai wanted to mold Zuko into the perfect heir, cold, emotionless, and aruthless firebender. And he would use her to do it. Ursa, who loved Zuko more than anything, knew she couldn't let that happen. She wanted Zuko to find his own path, his own way. And if own his own he turned out to be what Ozai wanted, then so be it. But she refused to manipulate her son just so that her husband could have his way.
And so, Ursa decided that in order to protect her son, she would have to leave the Fire Nation and never return. Even if it meant hurting her son, in the end it would be worth it. But she couldn't leave without saying goodbye to him. Knowing that even though she has tried not to, she is hurting her son, she leaves him with the words "Everything I've done, I'vedone to protect you. Remember this Zuko...no matter how things may seem to change, never forget who you are." hoping that he will understand some day. She obviously thinks she'll never get the chance to explain.
The most likely reason Ursa leaves at the footsteps(that fits into my theory at least) is that these are the footsteps of whoever is helping her escape, because obviously leaving a well-guarded palace in the middle of the night completely undetected is no easy task to do by yourself. While there is reason to believe that it could be a guard and so she must rush, it is more likely to be whoever is helping her, because of her unpanicked look at the sound of them.** If it were a guard, she would most likely have panicked and left the room in a hurry, as opposed to knowing she is merely sad she has to go.
It is the next day when Zuko asks his father where his mother is, and I believe that the reason he doesn't answer is because he is angry, and is reminded by Zuko's presence that he can never fully achieve what he had wanted to, with his wife gone. The look on his face when his father doesn't answer him is most likely one of shock and fear, because he's realized for once that Azula had been telling the truth: No one really does know where his mother is. Imagine that, loosing the only person in your family you'd ever liked.
I also think that later on in the series (around 3 or 4) Ursa will show up, and will in some way play a crucial role in defeating the Fire Nation. Personally, I think she's going to be teaching him the BASICS on firebending. You know, breath deeply, concentrate, the stuff he couldn't get while trying to learn it under Jeong Jeong. I have a feeling Ursa would be a very good teacher. Kind and wise, a little like Iroh.
*I say exile because of how we all know that Azula is an evil, sadistic, most likely schizophrenic, little brat. She jumps at any chance available, and some that even aren't to torture her older brother. She could have easily exaggerated. By no means would she have thought up the whole thing by herself, but she could have easily twisted it to better suit what she wanted to accomplish.
PS: My favorite thing about this episode is how when Zuko declares who he is, he says his mother's name first, emphasising the great love he had/has for her.
