What do correlation functions compute in CFT?  I would like to understand what correlation functions compute in Conformal Field Theory in mathematics. Let me begin with basic definitions. We define a free boson field $\phi(z)$ as a formal power series
$$
\phi(z)=q+a_0\log(z)-\sum_{n\ne0}\frac{a_n}{n}z^{-n},
$$
where $q,a_n$ for $n\in \mathbb{Z}$ are operators satisfying relations
$$
[a_n,q]=\delta_{n,0}, \ \ \ [a_m,a_n]=m\delta_{m+n,0}. 
$$
Here these operators acts on the $space$ generated by $vacuum$ vector $|0\rangle$ with properties 
$$
a_n |0\rangle=0\ (n\ge0), \ \ \langle0|q=\langle0|a_n=0\ (n<0), 
$$
where $\langle0|$ is the dual of the vacuum, i.e. $\langle0|0\rangle=1$. 
We then define the current $J(z)$ of $\phi(z)$ as a formal derivative
$$
J(z)=\sum_{n\in \mathbb{Z}}a_nz^{-n-1}. 
$$

Here is my question. In CFT one is interested in $correlation$ $functions$ such as
  $$
\langle0|J(z_1)J(z_2)|0\rangle, \ \ \langle0|J(z_1)J(z_2)J(z_3)J(z_4)|0\rangle
$$
  What do they compute in this context? 

In physics, correlation function like $\langle0|\phi(x)\phi(y)|0\rangle$ computes the possibility of the field $\phi(x)$ to become $\phi(y)$ etc where $x,y$ are coordinate in for example Minkowski space. How should one interprete the correlation function in our case? 
 A: I'm not sure exactly what kind of information you want, and CFT is an enormous subject, but here is some information
on the physical interpretation of the complex coordinates and correlation functions along with an example of their
mathematical interpretation in a special CFT. 
An ordinary refrigerator magnet contains a ferromagnetic material. The atoms in such a material have electrons which act as tiny magnets and they have an interaction between them (of quantum mechanical origin) which makes the spin/magnetic moments of the
electrons have lower energy when they are aligned.  This alignment produces a macroscopic magnetic field. I am simplifying here, because in real materials this interaction only operates over short distances and one actually forms domains of aligned spins with
the domains oriented randomly and a net magnetization is produced only by subjecting the material to an external magnetic field
which aligns the domains.
If one heats up such a ferromagnet then at some temperature the random thermal motion overcomes the tendency to align and
the macroscopic magnetic field goes to zero. The transition point between the state with net magnetization and the state with zero
magnetization is known as a second order critical point. The behavior of phase transitions  between different states (magnetized vs. unmagnetized, or water vs. ice or liquid vs. gas etc. ) has been one of the central topics in condensed matter physics for many years.  Various simplified models have been invented to try to understand such behavior analytically. The simplest of these is the
Ising model consisting of spins which take the value $\pm 1$ living on a two-dimensional lattice with nearest neighbor interactions between spins. There are many more complicated models, one of these is known as the Gaussian model because of the Gaussian weight used to define the probability distribution for spins. These models at the critical point have fluctuations on all length scales and one can take a continuum limit. This continuum limit is a conformal field theory. If the model is defined in two spatial dimensions then the continuum limit is a two-dimensional conformal field theory. The continuum limit of the Gaussian model is a conformal field theory with $c=1$ equivalent to the $c=1$ conformal field theory you have defined. The continuum limit of the Ising model is a $c=1/2$ conformal field theory consisting of a free Majorna fermion. The physical interpretation of the complex coordinates is simply that they are complex coordinates in the real two-dimensional plane describing the physical space on which the model is defined. The correlation functions measure the correlations between the microscopic spins at different spatial points at the critical point.
In two-dimensional conformal field theory the dependence on the coordinates of both the two-point and three-point correlation functions
is completely determined by conformal invariance, so the only interesting information is in the numerical coefficients appearing in
these correlation functions. For $c=1$ CFT there isn't any terribly interesting information in these correlation functions. However
for other CFT's there is more interesting information, including some of purely mathematical interest. For example, Frenkel, Lepowsky and Meurman constructed a $c=24$ CFT which has the Monster sporadic group as its automorphism group. This CFT has
196884 dimension 2 fields and the three point correlation function of these dimension 2 fields can be used to compute the
structure constants of the Griess algebra which was used in the original construction of the Monster. 
A: If you are happy with the interpretation you give at the bottom of your question for the correlation function $G_2(x,y)=\langle0|\phi(x)\phi(y)|0\rangle$ for a quantum field on Minkowski space, then it is not hard to relate that to the correlation of the CFT fields $\phi(z)$ or $J(z)$ that you are interested in. Namely, if the Minkowski correlation functions $G_n(x_1,\ldots,x_n) = \langle0|\phi(x_1)\cdots\phi(x_n)|0\rangle$ are real analytic in $x$ and $y$ (away from the diagonal, where $x_i=x_j$, for some $i\ne j$). The time coordinate $t_k$ of each point $x_k$ can then be analytically continued to the imaginary axis $t_k \to i\tau_k$, converting the Minkowski space point $x_k = (t_k,\xi_k)$ to a point on the complex plane $z_k = \xi_k + i\tau_k$. The same analytic continuation converts derivatives along left and right null directions, $\partial_\pm = \partial_\xi \pm \partial_t$, become the holomorphic and antiholomorphic differentials $\partial/\partial z$ and $\partial/\partial \bar{z}$ (matching which becomes which left as an exercise).
In Minkowski space, the fields $\phi(x)$ usually obey the wave equation $(\partial_\xi^2 - \partial_t^2)\phi(x) = 0$. This means that it has a decomposition into the left- and right-moving fields $\phi(x) = \phi_+(x) + \phi_-(x)$, which are annihilated by the respective differential operators $\partial_+$ or $\partial_-$. After analytic continuation (which need only be considered for the correlation functions rather than the fields themselves), the field can be thought to be split into holomorphic and antiholomorphic parts, $\phi(z,\bar{z}) = \phi(z) + \bar{\phi}(\bar{z})$. This is how the field $\phi(z)$ that you wrote down comes about. The definition you gave is essentially a shortcut through the above analytic continuation arguments.
So the correlation functions you are interested in are analytic continuations of correlations of products of left- or right-moving Minkowski quantum fields.
