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in the book Irena Peeva "Graded Syzygies", I saw that Polynomial ring has generic coordinates. But in this book, It is not defined anywhere. I looked for it but did not see a clear definition. I want to know the definition of generic coordinates.

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Very likely they are just considering the affine space and the associated polynomial ring $k[X_{1},X_{2}...X_{n}]$. In algebraic geometry it is customary to identity the set of maps from a variety as $R_{X}=k[X_{1},X_{2}...X_{n}]/I(X)$. You can consult serious algebraic geometers on this, for it is not my field and "Graded Syzygies" suggest some kind of commutative/homological algebra is involved.