If I have not made an error typing your code

jacobi[a0_,b0_,p0_,max_]:=Module[{a=N[a0],b=N[b0],i,j,k=0,n=Length[p0],p=p0,pold=p0},
  Print["P"o,"=",p];
  While[k<max,
    For[i=1,i<=n,i++,
      p[[i]]=1/a[[i,i]](b[[i]]+a[[i,i]]pold[[i]]-Sum[a[[i,j]]pold[[j]],{j,1,n}])];
    Print["p"k+1,"=",p];
    pold=p;
    k=k+1];
  Return[p];];
a={{2,8,3,1},{0,2,-1,4},{7,-2,1,2},{-1,0,5,2}};
b={-2,4,3,5};
vars={"x1","x2","x3","x4","x5","x6","x7","x8","x9"};
Print["Denklem Sistemi Cozumu"];
Print[MatrixForm[a],MatrixForm[vars],"=",MatrixForm[b]];
p={1,1,1,1,1,1,1,1,1};
x=jacobi[a,b,p,15]

then running that instantly displays

Denklem Sistemi Cozumu
<<<your matrix expression>>
Po={1,1,1,1,1,1,1,1,1}
Part 5  of {2, 8, 3, 1}  does not exist
Part 6  of {2, 8, 3, 1}  does not exist
Part 7  of {2, 8, 3, 1}  does not exist
Further output of Part::partw will be suppressed during this calculation

followed by a large expression with subscripts which cannot be evaluated because they are out of bounds.

Can you compare this with what you have and see if you can find the source of the difference?