Question:

Suppose that V is a normal variety, and Z is a principal divisor so a closed subvariety of V with codimension 1. Then how do we construct the valuation ring OZ？

Here describe the construction: take an open affine U=Spec R such that U intersection Z is nonempty and indeed a maximal proper closed subset of U, then U int Z corresponds to a minimal prime ideal of R, and so Rp is a normal ring with unique maximal pRp, and it is a DVR. We define OZ to be Rp. More intrinsically we can define OZ to be the set of rational functions on V that are defined over an open U such that U int V is not empty.

Since V is normal, we have R=OX(U)=Γ(U,O)R=OX(U)=Γ(U,O) is integrally closed， but why Rp is a discrete valuation ring. Moreover why is it important that p being a minimal ideal?

Answer:

This is basically pure algebra. By a theorem in chapter 9 of Atiyah Macdonald (I think Proposition 9.2 or 9.3):

 a Noetherian local domain of dimension 1 is a DVR iff it is integrally closed.

So in your case above we are given a minimal prime p (which is necessarily of height 1) and hence Rp is one-dimensional (basically because

 height of a prime = dimension of localization at this prime). Since Rp is integrally closed by the proposition above we have that Rp is a DVR.