the principal part
When you split a Laurent series into its upward-counting half and its downward-counting half, the downward half — the terms with negative powers — is called the principal part. The name is apt: this is the part that carries all the bad news. If the principal part is empty the function is secretly tame at z_0; the richer the principal part, the worse the singularity.
Precisely, the principal part of the Laurent series sum a_n (z - z_0)^n is the sub-sum over n less than 0, that is a_{-1}/(z - z_0) + a_{-2}/(z - z_0)^2 + a_{-3}/(z - z_0)^3 + ... — every term that grows without bound as z approaches z_0. The leftover, the n greater than or equal to 0 terms, is the analytic part, which stays finite and smooth at z_0. So Laurent series = principal part + analytic part, with the analytic part holomorphic on the whole disk and the principal part holomorphic everywhere EXCEPT at z_0.
The principal part is the official classifier of singularities. No negative terms (principal part is zero) means a removable singularity; finitely many negative terms, stopping at (z - z_0)^(-m), means a pole of order m; infinitely many negative terms means an essential singularity. Reading the principal part is therefore the first move whenever you meet an isolated singularity.
For f(z) = (cos z)/z^2 near 0, cos z = 1 - z^2/2 + z^4/24 - ..., so f = 1/z^2 - 1/2 + z^2/24 - .... The principal part is just 1/z^2 (here a_{-2} = 1, a_{-1} = 0), so f has a pole of order 2 at 0.
The principal part 1/z^2 names the singularity: a double pole, with no 1/z term at all.
The principal part depends on which annulus you expand in; the classification above uses the principal part on the INNERMOST punctured disk 0 < |z - z_0| < r, which is the only one that describes the singularity at z_0 itself.