the order of a zero or pole
When a function vanishes or blows up at a point, you want to know HOW STRONGLY. z vanishes 'once' at 0, but z^3 vanishes 'three times as hard' — it flattens out more. Likewise 1/z has a 'simple' blow-up while 1/z^4 explodes far faster. The order is the single integer that measures exactly this strength of vanishing or blowing up, with a clean sign convention: positive for zeros, negative for poles.
Precisely, let f be meromorphic near a point p on a Riemann surface, not identically zero, and pick a local coordinate z with z(p) = 0. Write the Laurent expansion f = sum of a_n z^n. The order ord_p(f) is the smallest n with a_n nonzero. So f = z^k g(z) with g holomorphic and g(0) nonzero gives ord = k > 0 (a zero of order k); f = g(z)/z^k gives ord = -k < 0 (a pole of order k); and ord = 0 means f is finite and nonzero at p. This integer does NOT depend on the choice of local coordinate, because a holomorphic change of coordinate w = w(z) with w'(0) nonzero preserves the leading power. To compute it in practice: factor out the highest power of z you can while keeping the rest finite and nonzero at p.
Why it matters: orders are the bookkeeping behind everything that follows. The sum over all p of ord_p(f) of the orders of any meromorphic function f on a COMPACT Riemann surface is ZERO — total zeros equal total poles, counted with order. (On the sphere this is the statement that a rational function has as many zeros as poles.) Packaging the orders of f as a formal sum gives its divisor. A common slip: order is defined by the LEADING term of the Laurent series, not the number of terms; z + z^2 has order 1 at 0 (the z term dominates), not order 2.
Take f(z) = (z^2 - 1)/z^3 on the sphere. At z = 1 and z = -1 it has simple zeros (order +1 each); at z = 0 it has a pole of order 3 (order -3); and at z = infinity, in the coordinate w = 1/z, f becomes w^3(1 - w^2)/1 ... working it out gives a zero of order 1. The orders sum to 1 + 1 - 3 + 1 = 0, as the compactness theorem demands.
On a compact surface the orders of any meromorphic function sum to zero — here 1 + 1 - 3 + 1.
Order is coordinate-independent only because we use HOLOMORPHIC coordinate changes (which have nonvanishing derivative). It is set by the lowest-degree nonzero Laurent term, not the count of terms; and don't forget to check the point at infinity on compact surfaces.