a primitive
On the real line, an antiderivative of f is a function F whose derivative is f, and once you have one, integration becomes trivial: you just evaluate F at the endpoints. The complex version is the same idea. A primitive of f on a domain is a holomorphic function F with F'(z) = f(z) everywhere on that domain. Having a primitive is the single most convenient thing that can happen to a contour integral.
The reason is the complex fundamental theorem of calculus: if F is a primitive of f on a domain, then the integral of f along any contour from a point z1 to a point z2 in that domain equals F(z2) - F(z1). The path drops out entirely — only the endpoints survive. In particular, around any closed loop (where z1 = z2) the integral is zero. So the existence of a primitive is a very strong property: it makes f's integral path-independent and kills every closed-loop integral.
But primitives are not guaranteed. The trouble case is the model integrand f(z) = 1/z on a region around the origin: it has no single-valued primitive there, because the natural candidate, log z, is multivalued — going once around the origin increases log z by 2 pi i. That is exactly why the integral of 1/z around the unit circle is 2 pi i rather than 0. A primitive exists when the domain has no holes that f's antiderivative cannot get around; the question of when that happens is the gateway to Cauchy's theorem.
f(z) = z^2 has primitive F(z) = z^3 / 3 on the whole plane. So the integral of z^2 along any contour from 0 to 1 + i is F(1+i) - F(0) = (1+i)^3 / 3 = (2i - 2)/3, regardless of the route taken.
With a primitive in hand, a contour integral collapses to a difference of endpoint values.
A primitive on a domain is unique up to an additive constant (two primitives differ by a function with zero derivative, hence a constant on each connected piece) — but its existence is a property of f and the shape of the domain together, not of f alone.