an isolated singularity
A singularity is a point where a function fails to be holomorphic — where it is undefined, blows up, or otherwise breaks the rules. It is called isolated when that point is the ONLY trouble nearby: the function is perfectly holomorphic on a small punctured disk all around it, with no other singularities crowding in. Think of a single pinprick in an otherwise smooth sheet, with breathing room on every side.
Precisely, z_0 is an isolated singularity of f if there is some radius r greater than 0 such that f is holomorphic on the punctured disk 0 < |z - z_0| < r — everywhere in that little disk except the centre. The point of demanding this room is that exactly here a Laurent series exists: f equals sum a_n (z - z_0)^n on the punctured disk, and from the principal part of THAT expansion you classify the singularity into one of three clean types — removable, a pole, or essential.
Many singularities are not isolated, and they fall outside this neat theory. The branch point at z_0 = 0 of log z is not isolated in the usual sense (you cannot puncture a disk and have log single-valued), and the function 1/sin(1/z) has poles at z = 1/(n pi) piling up toward 0, so 0 is a singularity that is a limit of other singularities — not isolated. The whole Laurent-series classification only applies to the isolated case; outside it you need other tools.
f(z) = 1/(z - 3) has a single isolated singularity at z_0 = 3: it is holomorphic on 0 < |z - 3| < r for any r, with nothing else broken nearby. By contrast f(z) = 1/sin(pi/z) has a non-isolated singularity at 0, since its poles at z = 1/n cluster there.
Isolated versus not: 1/(z-3) has elbow room around its singularity; 1/sin(pi/z) does not at 0.
Being isolated is a precondition for the three-way classification, not a guarantee the function is nice — an essential singularity is isolated yet wildly badly behaved (Casorati-Weierstrass and Picard).