the order of a pole
Not all poles blow up equally fast. 1/(z - z_0) goes to infinity, but 1/(z - z_0)^2 races there far faster, and 1/(z - z_0)^5 faster still. The order of a pole is the single whole number that measures how violently the function blows up — how steep the spike is. A pole of order 1 is called simple; orders 2, 3, ... are double, triple, and so on.
Formally, the order is the size m of the deepest negative term in the Laurent series: if f(z) = a_{-m}/(z - z_0)^m + ... + a_{-1}/(z - z_0) + (analytic part) with a_{-m} not zero, the pole has order m. There is a tidy equivalent test that avoids series: m is the smallest positive integer for which the product (z - z_0)^m times f(z) has a removable singularity at z_0 — that is, stays bounded (in fact tends to a nonzero limit). So you find the order by asking how many factors of (z - z_0) you must multiply in to tame the blow-up. A third route: if f = g/h with g, h holomorphic, g(z_0) not zero, then the order of the pole of f equals the order of the zero of h at z_0.
Knowing the order is the practical first step before computing a residue, because the formula you use depends on it (a simple pole uses a one-line limit; a higher-order pole needs derivatives). It also obeys clean arithmetic: multiplying functions adds pole orders, and a zero of order k in the numerator cancels k units of pole order in the denominator. This bookkeeping of orders is what makes meromorphic functions so manageable.
For f(z) = 1/(z^2 sin z) at z_0 = 0: sin z has a simple zero at 0 (sin z is about z), so 1/sin z has a simple pole, and the extra z^2 in front makes 1/(z^2 sin z) blow up like 1/z^3 — a pole of order 3. Check: z^3 f(z) = z/sin z tends to 1, a nonzero limit, confirming order 3.
Counting orders: a double factor times a simple zero of sin gives a pole of order three.
The order is defined only for poles — it is a finite positive integer; an essential singularity has 'infinitely many' negative terms and so no finite order, and a removable singularity has order zero in the sense of no negative terms at all.