A complex function (i.e. a function f: C → C, where C is the complex plane) is called holomorphic at a point z if it has a complex derivative
f'(z) = lim|h|→0 (f(z+h)-f(z))/h.
If the derivative is defined on all points in a
domain, f is said to be holomorphic on that domain. The definition exactly parallels that of
differentiablity on the
real number line.
However, it turns out that if f is holomorphic, then its partial derivatives with respect to real and imaginary parts must satisfy the Cauchy-Riemann equations. This connection in turn leads to f' itself being differentiable (as a real function R2→R2, where R2 is identified with C in the usual manner), and holomorphic. So such an f is infinitely differentiable! Furthermore, it turns out that the Taylor series for f near any point converge to f -- f is analytic.
This is much more than you get on the real line. There, if f is differentiable at a point, f' itself needn't be (consider, e.g., an indefinite integral of the function abs(x)=|x|). And even if f is infinitely differentiable, it still might not be analytic! (For an example, consider exp(-1/x^2) (and 0 at 0), which has all derivatives 0 at 0, but is positive for all x≠0!)
This extreme "rigidity" of holomorphic functions makes complex analysis very different from real analysis.