the mean-value property
Stand at any interior point of a holomorphic function and draw a small circle around yourself. The mean-value property says the value at your center is exactly the average of the values around the circle. The function does not bulge up or sag down at the center relative to its surroundings; the center is always the perfect average of the ring.
In symbols: if f is holomorphic on and inside the circle of radius r centered at z_0, then f(z_0) = (1 / (2 pi)) times the integral from 0 to 2 pi of f(z_0 + r e^(i theta)) dtheta. This is Cauchy's integral formula specialized to a circle and rewritten: parametrize the contour as z = z_0 + r e^(i theta), substitute into f(z_0) = (1 / (2 pi i)) times the integral over the circle of f(z) / (z - z_0) dz, and the dz and the kernel combine so the radius and the i cancel, leaving a plain angular average. The center value is literally the mean of the boundary values.
The same averaging holds for the real and imaginary parts separately, which are harmonic functions, so harmonic functions also satisfy the mean-value property. From this single fact flow the maximum and minimum principles: if the center always equals the average, it can never strictly exceed everything around it, so |f| and harmonic functions cannot have strict interior maxima. The property is also the discrete heartbeat behind why solutions of Laplace's equation smooth out and have no isolated bumps.
For f(z) = z and z_0 = 0, the average of f around the circle |z| = r is (1 / (2 pi)) times the integral of r e^(i theta) dtheta from 0 to 2 pi, which is 0 — exactly f(0), as the property promises.
The center value equals the angular average over any surrounding circle.
The mean-value property characterizes harmonic (and holomorphic) functions: a continuous function with this property at every point and every small radius is automatically harmonic — it is not merely a side effect.