Let \(f\) be holomorphic on a punctured disc \(D_r(z_0)\setminus\{z_0\}\). The point \(z_0\) is a removable singularity of \(f\) if there exist \(0<\rho\le r\) and a holomorphic function \(g\colon D_\rho(z_0)\to\CC\) with \(f=g\) on \(D_\rho(z_0)\setminus\{z_0\}\).
The point \(z_0\) is a removable singularity of \(f\) if and only if there are an integer \(n\le0\) and a function \(h\) holomorphic on a neighborhood of \(z_0\) with \(f(z)=(z-z_0)^{-n}h(z)\) for \(z\neq z_0\) near \(z_0\).
If \(z_0\) is removable, take \(n=0\) and \(h=g\). Conversely, \((z-z_0)^{-n}h(z)\) is holomorphic near \(z_0\) when \(-n\ge0\).
The factorization \(f=(z-z_0)^{-n}h\) with \(n\ge1\) defines a pole of order $n$; allowing \(n=0\) in it gives exactly the removable singularities, which is the sense of the phrase “pole of order zero”.