Removable singularity
An isolated singularity where f extends to be analytic there too, once assigned the right value.
An isolated singularity z_0 of f is removable if f can be redefined (or defined, if it wasn't already) at z_0 so that the result is analytic at z_0 -- equivalently, the Laurent series of f around z_0 has an all-zero principal part, and equivalently, f stays bounded on some punctured disk around z_0. A removable singularity is, in a real sense, not a genuine singularity at all: it is a gap in the definition of an otherwise perfectly analytic function.
Examples
f(z)=\dfrac{\sin z}{z} has a removable singularity at z=0: defining f(0):=1 makes f analytic there, matching the Laurent series 1-\tfrac{z^2}{6}+\cdots.
f(z)=\dfrac{z^2-1}{z-1} has a removable singularity at z=1, since f(z)=z+1 for z\ne 1 and setting f(1):=2 removes it.