Isolated singularity

A point where $f$ fails to be analytic, but $f$ is analytic on some punctured disk around it.

A point $z_0$ is an isolated singularity of $f$ if $f$ is analytic on some punctured disk $0<|z-z_0|<r$ but not analytic at $z_0$ itself (either undefined there, or defined but failing to be differentiable). "Isolated" rules out singularities that pile up arbitrarily close to $z_0$ : every isolated singularity has breathing room around it where $f$ is perfectly well-behaved, which is exactly what makes a Laurent series expansion around it possible.

Examples

$z_0=0$ is an isolated singularity of $f(z)=1/z^2$ , since $f$ is analytic on $0<|z|<\infty$ .
$f(z)=1/\sin(1/z)$ has singularities at $z=1/(n\pi)$ accumulating at $0$ , so $z_0=0$ is a non-isolated singular point: every punctured disk around it contains infinitely many other singularities.