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.