What was the alternative world in Sucker Punch?

This is actually two questions. Muad'Dib is correct that the fantasy world (the brothel) is her way of coping with the real world (the asylum).

Baby Doll (I assume you mean her; Blondie was a supporting character) dances in the fantasy world, which transports the audience to a third level, the one with the fighting and fantastical elements. I believe this to be some sort of manifestation of her dance, as if her dancing would provide such imagery to the person observing her. When the train heist mission is botched, so is Baby Doll's dance.

It's all reads a bit like Inception, only not as intelligently done.

It's all in her head. It is her way of coping with the insanity of the real world situation she is in. The events that take place in her alternate reality echo the events taking place in the real world.

Actually I have another interpretation.

The whole movie the Wise Man (played by Scott Glenn) appears in the fantasy world, but at the end of the movie only Sweet Pea meets him on the bus, because Baby Doll dies before that scene. She can't insert him on her fantasy world because they never meet each other.

So, all we see, even when Baby Doll dances and there's a crazy version of the real world being shown, that's Sweet Pea telling the story to someone after all happened.