a divisor
Meromorphic functions and differentials carry their data in their zeros and poles: WHERE they vanish or blow up, and HOW HARD. A divisor is the simple bookkeeping device that records exactly this — a finite list of points each tagged with an integer (positive for a zero, negative for a pole, the integer telling the order). It turns 'this function vanishes to order 3 here and has a double pole there' into one clean algebraic object you can add and subtract.
Precisely, a divisor on a Riemann surface X is a formal finite integer combination of points: D = sum of n_p [p], where each n_p is an integer and only finitely many are nonzero. Divisors form a free abelian group (you just add the coefficients point by point). Its degree is the sum of the coefficients, deg(D) = sum of n_p — a single integer. Two key sources: any meromorphic function f gives its principal divisor div(f) = sum over p of ord_p(f) [p] (zeros minus poles); on a compact surface deg(div(f)) = 0 always, since zeros and poles balance. A divisor with all n_p >= 0 is called effective (written D >= 0); it records a configuration of points with multiplicities and nothing negative.
Why it matters: divisors are the language of the whole Riemann-Roch circle of ideas. Each divisor D defines a vector space L(D) of meromorphic functions whose poles are bounded by D (and which may vanish more), and Riemann-Roch computes its dimension. Divisors that differ by a principal divisor are linearly equivalent and define the same geometry; classes of divisors under this relation form the Picard group, and the degree-zero part is the Jacobian. Honesty point: a divisor is just FORMAL data — a finite labelled set of points — it is not itself a function. The art is asking which divisors actually arise as div(f) or as the zeros of a section, which is exactly what Abel's theorem and Riemann-Roch answer.
On the sphere take f(z) = (z - 1)^2 / (z(z + 3)). Its divisor is div(f) = 2[1] - [0] - [-3] - 0[infinity], and checking degree: 2 - 1 - 1 = 0, confirming the principal-divisor rule. The effective divisor 2[1] alone records 'a double point at z = 1' with no poles.
A principal divisor div(f) always has degree 0 on a compact surface — zeros exactly balance poles.
A divisor is formal data, not a function: writing D = sum n_p[p] says nothing about whether a function with those zeros and poles exists. Deciding which divisors are principal (degree 0 is necessary but NOT sufficient when g >= 1) is precisely Abel's theorem.