Answer
The graph is a circle with a center at $(1,0)$ and a radius of $r=2$. See details below.
Work Step by Step
$\begin{align*}
\vert z-1\vert &= 2 \\
\vert (x+iy)-1\vert &= 2 \\
\vert (x-1) +iy\vert &= 2 \\
\sqrt{(x-1)^2 + y^2} &= 2 \\
(x-1)^2 + y^2 &= 4
\end{align*}$
The graph is a circle with a center at $(1,0)$ and a radius of $r=2$.
