How to understand that correlators measure physical correlation? Background In physics, we always come across $n$-point correlators (e.g. 2-point correlators). For instance, in phase transitions, one is interested in finding the correlation function between order parameters $\mathscr{m}$ at two different points $\langle \mathscr{m}(\textbf{x})\mathscr{m}(\textbf{x}^\prime)\rangle$. In cosmology, deals with the temperature-temperature correlator $\langle T(\textbf{n})T(\textbf{n}^\prime)\rangle$ between two locations in the CMB sky. Examples can also be quoted from optics, quantum field theory etc.
Questions
(A) Why do the mathematical objects such as $\langle A(x) A(x^\prime)\rangle$ give a measure of any sort of physical correlation? As far as I'm concerned, the "correlation function" is just a name. How and why should it be related to actual correlation is not clear to me.
(B) What am I supposed to understand, if all (or odd or even) $n$-point correlation functions are zero?
(C) Can we physically interpret the meaning of the correlators and the physical information they carry? 
 A: (A) As LonelyProf has answered above the quantity $\langle A(t) A(t') \rangle$ where the $\langle \rangle$ denotes statistical average is dimension-ful and hard to interpret.  The dimensionless correlation coefficient:
$ \rho \equiv \frac{\langle A(t) A(t') \rangle}{\sqrt{\langle A(t) A(t) \rangle}\langle A(t') A(t') \rangle}$
is a number $\in [-1, 1]$.  This numbers quantifies how "the same" the values of A are at t and t'.  In particular, on this scale, -1 means completely opposite (everytime you see A at t, you can conclude -A happens at t') and 1 means completely the same.  Hence why we call the quantity in braket loosely "correlators". Note that we assume for simplicity $\langle A \rangle$ is 0 $\forall t$.
B. For starters, if all odd ("self") correlators at a given time is zero, it means that the probability distribution of A at that time is symmetric around 0.  If all even "self" correlators are 0, it means the probability distribution for A at that time is odd.  You can reconstruct a probability distribution by using the inverse fourier transform:
$ \frac{\partial^n}{\partial (-ik)^n} \int p(x) e^{-ikx} dx = \langle x^n \rangle$
$p(x)  = \frac{1} {2 \pi} \int dx e^{ikx} \sum_n \frac{ (-ik)^n \langle x^n \rangle }{n!}$
in general so knowing the symmetry of the correlators tells you about the symmetry of the fourier transform of a distribution.
C.  See A for the meaning of correlation.  
You mentioned a new word, "information."  There is a technical meaning to that word discovered by Shannon.  I'm not sure if that's what you want to get at.  Anyways, given a probability distribution for some quantity (in this case could be $p(A(t)$)), Shannon quantified the amount of information you get when you measure A(t) to be some $a_0$.  
This is called the Shannon entropy, and is just 
$\langle log_2(p) \rangle$
for the discrete case ( $p(A(t))$ is a bunch of dirac deltas).  
For the continuous case, this quantity is ill defined but you can compute the "differential entropy" analog.  Part B told you how to reconstruct a probability distribution from its "correlators".  The only tricky part in physics is usually that our probability distributions are in many variables (uncountably many in the case of a stochastic functions A(t) in fact). 
