In complex analysis, we're constantly faced with problems about the analyticity of a function, on which many theorems are developed. I of course know a bunch of formulas and theorems, but could not have any intuitions as I do in realnumbered world. So, what, intuitively, does the property of analyticity mean? Could you give some hint of how to relate this property to real analysis?

$\begingroup$ What is complex analysis? $\endgroup$ – Alex Degtyarev Nov 16 '14 at 12:43

3$\begingroup$ perhaps the comments and answers at mathoverflow.net/questions/3819/… may be helpful $\endgroup$ – Yemon Choi Nov 16 '14 at 12:46

$\begingroup$ @AlexDegtyarev the analysis of functions of complex variables $\endgroup$ – Austin at CA Nov 16 '14 at 12:47

$\begingroup$ One complex variable? Or many? $\endgroup$ – Gerald Edgar Nov 16 '14 at 12:52

$\begingroup$ @GeraldEdgar one $\endgroup$ – Austin at CA Nov 16 '14 at 12:54
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 padic 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 reexpanded 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 2nd 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).

1$\begingroup$ For extra visibility to those reading: an old question with similar answers mathoverflow.net/questions/3819/… $\endgroup$ – Yemon Choi Nov 16 '14 at 15:06
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 CauchyRiemann equation(s).
Real analytic functions lack such characterizations. Also, the celebrated Cauchy residue formula is a manifestation of the fact that the Cauchy kernel
$$\frac{1}{\pi\boldsymbol{i} z} $$

1$\begingroup$ Yes. And the crucial point here is that the CR system is elliptic. Therefore all its solutions are analytic, in the sense that they can be expanded into a Taylor series:) $\endgroup$ – Alexandre Eremenko Nov 16 '14 at 16:22