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Hi, how is it defined the normalization of an algebraic stack $A$ inside another algebraic stack $B$. If you do not want to write the answer could you give to me some reference? Thank you

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I don't know why you want an "ambient" stack, anyway in these notes math.upenn.edu/~kresch/teaching/AlgStacks.pdf pages 309-310, there is something about normalizations. To construct it, I think that you can just take a groupoid presentation of your stack (say it is reduced and irreducible to stay safe), take the normalization of the atlas and pullback to it the whole groupoid, and then take the associated stack of the new groupoid. –  Mattia Talpo Jan 28 '11 at 11:55
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To define the normalization of $A$ in the absolute setting (i.e. no stack $``B"$), one can use 2-universal property. Note that since normality is local for the smooth topology, we know what a normal algebraic stack means: it is normal if it has a presentation which is a normal scheme. To construct it we follow Mattia's comment.

For the relative setting, I don't quite understand, but I guess you mean there is a given morphism $B\to A.$ Do you want it to be representable?

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