First of all, I must start by explaining that this is only a personal interpretation of the meaning of the song's lyrics. Noel Gallagher, who wrote the song himself, stated it was not about anything. He wrote it while under the influence and the character "Sally" is not a real person, and that the line "So Sally can wait" was conceived when Noel was just goofing around and his brother Liam thought he was singing those words. They thought it sounded good and so they turned it into a song. For the full story, I refer you to Wikipedia.
I guess the song isn't deep then, or wasn't intended to be, but nevertheless it is an amazing piece of poetry from which much meaning can be derived.
So my personal interpretation of the song. The true meaning of the song is to say that everybody goes through bad breakups and that the best thing to do us just to move on, continue with your life while learning from your mistakes and just letting the bad times go. The specific story in Don't Look Back In Anger is about a breakup between the singer and his ex girlfriend Sally. In the song, the character Sally is anger and bitter. She wants to get back together with the singer, or she wants to make him burn to make up for what he did to her. The singer's attitude is relaxed. He feels like it's mostly the fault of the rock and roll lifestyle he leads, and wants to just forget the whole thing and move on. The two had obviously been a close couple until things went bad between them, and the singer- we'll call him our male protagonist- called it off. His tone is forgiving though- he doesn't really care anymore about what happened between himself and Sally, he's done and moved on, and the purpose of the song is that he is telling her to do the same thing.
The song opens with what is to me a clear drug reference:
Slip inside the eye of your mind
Don't you know you might find
A better place to play
Most likely a reference to hallucinogens, this line is the singer telling Sally to go to a different part of her mind, and to think about things a different way, using drugs if she needs to. The next line
You said that you'd never been
All the things that you've seen
Will slowly fade away
Is Sally stating that she has never done drugs before- we're probably talking about acid tripping here. Then the singer encourages her some more, telling her that all the bad things that have happened between the two of them will fade from her mind and she will be able to not think about it for a little while, which would benefit her.
Gonna start a revolution from my bed
Cause you said the brains I had went to my head
This line here is a John Lennon reference. Lennon actually said almost these exact words himself once in an interview, and Gallagher tossed the lines in as an homage to his hero. The words almost certainly refer to Lennon's bed-ins in the late 60s. Beatles influences and references can be commonly heard in several Oasis songs, and the opening piano chords in this song are strongly reminiscent of "Imagine."
Step outside the summertime's in bloom
In this line our singer tells Sally that everything is okay, and that she just needs to open her eyes and see that everything really is alright.
Stand up beside the fireplace
Take that look from off your face
You ain't never gonna burn my heart out
Here he tells Sally to stop wasting her time. She is putting to much effort into trying to hurt him and to get back at him. He thinks she is taking the situation too seriously, and he's telling her to stop, because he's never going to feel bad about what happened.
The second verse begins with what might be another drug reference-
Take me to the place where you go
Where nobody knows
If it's night or day
He may again be implying that they should use drugs to alter their mental state so they can either cool down and look at their situation a different way, or just to relax and forget about their lives for a while.
Don't put your life in the hands
Of a rock and roll band
Who will throw it all away
This is the saddest and most humane part of the song. Here the singer accepts much responsibility for what happened between him and Sally. He admits that his profession caused him to do things in their relationship he may not have done otherwise- maybe seeing other girls while touring, or spending much time away from the house, or partying excessively, all common habits of rock and roll bands which greatly hurt the ones- particularly the women- they love. He tells her here that a relationship with a musician like this will only do her harm, and he begs her next time to choose a more stable partner, not to put all her heart into a musician who won't treat her as well as she deserves.
Then the chorus:
And so Sally can wait
She knows it's too late as she's walking on by
Her soul slips away
But don't look back in anger I heard you say
Here he says that Sally knows in her heart that things have gone bad between them and that they are never going to be together again. As they pass each other in silence, with no feelings from him for her anymore, her spirits fade because she realizes there is no hope left. He simply tells her that she should not look back in anger at what happened between them- the "you" would be the conscience of the main character- and to move on.
At the end of the song, the chorus ends with;
But don't look back in anger
At least not today
What is meant by this line in the song is that maybe one day the time will be right to look back on the situation and feel angry about it, but that time is not now, and until then, she should forget it and be happy with life. It isn't a situation to be worried about now. Maybe in the future, things will be bad and you can think about this angrily, but right now, all is well. Right now, Sally, don't worry about it. Put it behind you and move on, and please don't look back in anger.