Neighborhood

Any open disk around a point, used as the standard notion of "nearby."

A neighborhood of a point $z_0$ is an open disk $|z-z_0|<\varepsilon$ centered there, for some radius $\varepsilon>0$ . The word doesn't commit to a particular radius; it's a stand-in for 'some open disk around $z_0$ , as small as you like,' which is exactly the flexibility needed to state definitions of limits, continuity, and openness without fixing a number in advance.

Examples

"Every neighborhood of $z_0$ contains points where $f$ is undefined" means: no matter how small a disk you draw around $z_0$ , it catches such a point.
A punctured neighborhood of $z_0$ is a neighborhood with $z_0$ itself removed.