convergence divergence - Riemann Zeta Function Analytic Continuation

I am struggling to understand how the analytic continuation of the Riemann Zeta function is derived to extend it to all complex values $z$ not equal to $1$, starting with the series which converges only for $Re(z)>1$. Can someone provide a relatively simple/intuitive explanation of how this is achieved? Also, I understand that analytic continuation is unique, so there is only one analytic continuation of the Riemann Zeta Function, right?