What does analyticity imply in complex analysis?

EDIT: 11.16.2014, 3:40pm ET. Let me incorporate Liviu Nicolaescu's answer. (I am not sure whether this is legitimate:-)

Analyticity is a property encountered not only in complex analysis: a function of a real variable (or of a p-adic variable) can be analytic. 'Analytic' really means that in a neighborhood of every point in the domain, the function can be expanded in a convergent Taylor series. This implies the uniqueness theorem: zeros of such a function (of one variable) are isolated, unless it is identically equal to zero. So analytic functions are rigid objects: if two of them are equal in a neighborhood of a point, then they are equal everywhere. This is the main point which is responsible for the dramatic distinction between the theory of analytic functions and the rest of analysis.

Some of the fundamental facts about analytic functions are the following:

a) if a power series is convergent at more than one point, then it converges in a whole neighborhood of a point (Abel), and

b) once you have ONE convergent power series, then it can be re-expanded about another point in a region of convergence into a new power series. (Weierstrass). These facts are independent on the number system used. These facts make possible the "analytic continuation" process, which has no analog in the rest of Analysis. (Of course analytic continuation is most interesting over the complex numbers).

On a unified exposition of analytic functions over any complete normed field, see Bourbaki, Elements de mathematiques, XXXIII, XXXIV. Varietes differentielles et analytiques.

The specific of complex analysis is that simple differentiability (which looks like a much weaker property) implies analyticity. This is one of the great miracles of complex numbers (See Roger Penrose, "The road to reality. A complete guide to the laws of the universe" for the other miracles).

The miracle consists in the fact that simple differentiability requirement implies not only real differentiability but also a PDE which is called CR. This PDE is ELLIPTIC (and autonomous). Solutions of autonomous elliptic PDE are analytic (=expand in a Taylor series). For general 2-nd order elliptic PDE this is a theorem of Bernstein.

Cauchy's kernel is the fundamental solution of this PDE. This is the essence of Cauchy's theory. So we have the Cauchy integral formula which expresses the function inside the region in terms of its boundary values.

One reason why this class of functions is important is that solutions of differential and many other functional equations are analytic under certain broad conditions. When Newton wanted to state his main discovery in Calculus in one sentence, he said roughly speaking the following: "I can evaluate any integral or solve any differential equation by substituting a power series with undetermined coefficients in it" (this is my own understanding of his coded message:-) See: Arnold on Newton's anagram For example, $y(z)=e^z$ is the unique solution of $y'=y, y(0)=1$. So it is analytic.

There are at least two other independent reasons why analytic functions are important: one has to do with harmonic analysis (generating functions, in particular), another with spectral theory (resolvent is analytic outside of the spectrum).

The complex analytic functions, those that admit power series expansions in one complex variable, can also be characterized as solutions of a certain elliptic partial differential equation, namely the Cauchy-Riemann equation(s).

Real analytic functions lack such characterizations. Also, the celebrated Cauchy residue formula is a manifestation of the fact that the Cauchy kernel

$$\frac{1}{\pi\boldsymbol{i} z} $$

is a fundamental solution of the Cauchy-Riemann operator.