a holomorphic differential
On the plane, dz is the basic object you integrate: 'integral of f dz' is the bread and butter of complex analysis. On a curved Riemann surface you cannot use a single global z, so the right object is a DIFFERENTIAL — something that locally reads f(z) dz and transforms correctly when you switch charts, so that integrating it along a path makes coordinate-free sense. A holomorphic differential is one with no poles at all: everywhere finite, the gentlest and most fundamental kind.
Precisely, a holomorphic 1-form on a Riemann surface X is an assignment, in each chart with coordinate z, of an expression f(z) dz with f holomorphic, such that on overlaps the expressions agree under the chain rule: if w = w(z) is the transition, then the w-chart's g(w) dw equals f(z) dz, i.e. g(w(z)) w'(z) = f(z). The 'dz' is not decoration — it is precisely what makes the object transform as a section of the cotangent bundle (the canonical line bundle), so its zeros and poles are intrinsic. 'Holomorphic' means f has no poles in any chart, so the differential has zeros but never poles; its divisor is effective. The grand fact: on a compact surface of genus g, the holomorphic 1-forms form a complex vector space of dimension EXACTLY g.
Why it matters: holomorphic differentials are the analytic incarnation of the genus and the engine of the deepest theorems. Their dimension g matches the topological genus (Hodge theory: g comes from the harmonic forms). A basis omega_1, ..., omega_g lets you integrate over loops to get periods, which assemble into the period lattice and the Jacobian, and lets you write the Abel-Jacobi map p -> (integral from p_0 to p of omega_1, ..., integral of omega_g). The canonical class is the divisor class of any meromorphic differential, holomorphic ones being the effective representatives. Honesty point: a holomorphic differential is NOT a holomorphic function times dz globally — there is no global z. On a compact surface there are no nonconstant holomorphic functions, yet there are g independent holomorphic 1-forms, precisely because the dz-transformation law is different from a function's.
On the torus C/L the form omega = dz descends to a global holomorphic 1-form: it has no zeros and no poles (consistent with deg K = 2g - 2 = 0 for g = 1), and it is the UNIQUE holomorphic differential up to scaling — matching the count g = 1. Integrating dz over the two basis loops gives the two periods that define the lattice L itself.
On a torus the only holomorphic differential (up to scale) is dz; its periods over the two loops generate the lattice.
A holomorphic 1-form is a cotangent-bundle section, not a function times a fixed dz; the dz transforms by the chain rule, which is why a compact surface with NO nonconstant holomorphic functions can still carry g independent holomorphic differentials.