The correlation function you wrote is a completely general correlation of two quantities,
$$\langle f(X) g(Y)\rangle$$
You just use the symbol $x'$ for $Y$ and the symbol $x+x'$ for $X$.
If the environment - the vacuum or the material - is translationally invariant, it means that its properties don't depend on overall translations. So if you change $X$ and $Y$ by the same amount, e.g. by $z$, the correlation function will not change.
Consequently, you may shift by $z=-Y=-x'$ which means that the new $Y$ will be zero. So
$$\langle f(X) g(Y)\rangle = \langle f(X-Y)g(0)\rangle = \langle f(x)g(0) \rangle$$
As you can see, for translationally symmetric systems, the correlation function only depends on the difference of the coordinates i.e. separation of the arguments of $f$ and $g$, which is equal to $x$ in your case.
So this should have explained the dependence on $x$ and $x'$.
Now, what is a correlator? Classically, it is some average over the probabilistic distribution
$$\langle S \rangle = \int D\phi\,\rho(\phi) S(\phi)$$
This holds for $S$ being the product of several quantities, too. The integral goes over all possible configurations of the physical system and $\rho(\phi)$ is the probability density of the particular configuration $\phi$.
In quantum mechanics, the correlation function is the expectation value in the actual state of the system - usually the ground state and/or a thermal state. For a ground state which is pure, we have
$$\langle \hat{S} \rangle = \langle 0 | \hat{S} | 0 \rangle$$
where the 0-ket-vector is the ground state, while for a thermal state expressed by a density matrix $\rho$, the correlation function is defined as
$$\langle \hat{S} \rangle = \mbox{Tr}\, (\hat{S}\hat{\rho})$$
Well, correlation functions are functions that know about the correlation of the physical quantities $f$ and $g$ at two points. If the correlation is zero, it looks like the two quantities are independent of each other. If the correlation is positive, it looks like the two quantities are likely to have the same sign; the more positive it is, the more they're correlated. They're correlated with the opposite signs if the correlation function is negative.
In quantum field theory, correlation functions of two operators - just like you wrote - is known as the propagator and it is the mathematical expression that replaces all internal lines of Feynman diagrams. It tells you what is the probability amplitude that the corresponding particle propagates from the point $x+x'$ to the point $x'$. It is usually nonzero inside the light cone only and depends on the difference of the coordinates only.
An exception to this is the Feynman Propagator in QED. It is nonzero outside the light cone as well, but invokes anti-particles, which cancel this nonzero contribution outside the light cone, and preserve causality.
Correlation functions involving an arbitrary positive number of operators are known as the Green's functions or $n$-point functions if a product of $n$ quantities is in between the brackets. In some sense, the $n$-point functions know everything about the calculable dynamical quantities describing the physical system. The fact that everything can be expanded into correlation functions is a generalization of the Taylor expansions to the case of infinitely many variables.
In particular, the scattering amplitude for $n$ external particles (the total number, including incoming and outgoing ones) may be calculated from the $n$-point functions. The Feynman diagrams mentioned previously are a method to do this calculation systematically: a complicated correlator may be rewritten into a function of the 2-point functions, the propagators, contracted with the interaction vertices.
There are many words to physically describe a correlation function in various contexts - such as the response functions etc. The idea is that you insert an impurity or a signal into $x'$, that's your $g(x')$, and you study how much the field $f(x+x')$ at point $x+x'$ is affected by the impurity $g(x')$.
Best Answer
The main distinction you want to make is between the Green function and the kernel. (I prefer the terminology "Green function" without the 's. Imagine a different name, say, Feynman. People would definitely say the Feynman function, not the Feynman's function. But I digress...)
Start with a differential operator, call it $L$. E.g., in the case of Laplace's equation, then $L$ is the Laplacian $L = \nabla^2$. Then, the Green function of $L$ is the solution of the inhomogenous differential equation $$ L_x G(x, x^\prime) = \delta(x - x^\prime)\,. $$ We'll talk about its boundary conditions later on. The kernel is a solution of the homogeneous equation $$ L_x K(x, x^\prime) = 0\,, $$ subject to a Dirichlet boundary condition $\lim_{x \rightarrow x^\prime}K(x,x^\prime) = \delta (x-x^\prime)$, or Neumann boundary condition $\lim_{x \rightarrow x^\prime} \partial K(x,x^\prime) = \delta(x-x^\prime)$.
So, how do we use them? The Green function solves linear differential equations with driving terms. $L_x u(x) = \rho(x)$ is solved by $$ u(x) = \int G(x,x^\prime)\rho(x^\prime)dx^\prime\,. $$ Whichever boundary conditions we what to impose on the solution $u$ specify the boundary conditions we impose on $G$. For example, a retarded Green function propagates influence strictly forward in time, so that $G(x,x^\prime) = 0$ whenever $x^0 < x^{\prime\,0}$. (The 0 here denotes the time coordinate.) One would use this if the boundary condition on $u$ was that $u(x) = 0$ far in the past, before the source term $\rho$ "turns on."
The kernel solves boundary value problems. Say we're solving the equation $L_x u(x) = 0$ on a manifold $M$, and specify $u$ on the boundary $\partial M$ to be $v$. Then, $$ u(x) = \int_{\partial M} K(x,x^\prime)v(x^\prime)dx^\prime\,. $$ In this case, we're using the kernel with Dirichlet boundary conditions.
For example, the heat kernel is the kernel of the heat equation, in which $$ L = \frac{\partial}{\partial t} - \nabla_{R^d}^2\,. $$ We can see that $$ K(x,t; x^\prime, t^\prime) = \frac{1}{[4\pi (t-t^\prime)]^{d/2}}\,e^{-|x-x^\prime|^2/4(t-t^\prime)}, $$ solves $L_{x,t} K(x,t;x^\prime,t^\prime) = 0$ and moreover satisfies $$ \lim_{t \rightarrow t^\prime} \, K(x,t;x^\prime,t^\prime) = \delta^{(d)}(x-x^\prime)\,. $$ (We must be careful to consider only $t > t^\prime$ and hence also take a directional limit.) Say you're given some shape $v(x)$ at time $0$ and want to "melt" is according to the heat equation. Then later on, this shape has become $$ u(x,t) = \int_{R^d} K(x,t;x^\prime,0)v(x^\prime)d^dx^\prime\,. $$ So in this case, the boundary was the time-slice at $t^\prime = 0$.
Now for the rest of them. Propagator is sometimes used to mean Green function, sometimes used to mean kernel. The Klein-Gordon propagator is a Green function, because it satisfies $L_x D(x,x^\prime) = \delta(x-x^\prime)$ for $L_x = \partial_x^2 + m^2$. The boundary conditions specify the difference between the retarded, advanced and Feynman propagators. (See? Not Feynman's propagator) In the case of a Klein-Gordon field, the retarded propagator is defined as $$ D_R(x,x^\prime) = \Theta(x^0 - x^{\prime\,0})\,\langle0| \varphi(x) \varphi(x^\prime) |0\rangle\, $$ where $\Theta(x) = 1$ for $x > 0$ and $= 0$ otherwise. The Wightman function is defined as $$ W(x,x^\prime) = \langle0| \varphi(x) \varphi(x^\prime) |0\rangle\,, $$ i.e. without the time ordering constraint. But guess what? It solves $L_x W(x,x^\prime) = 0$. It's a kernel. The difference is that $\Theta$ out front, which becomes a Dirac $\delta$ upon taking one time derivative. If one uses the kernel with Neumann boundary conditions on a time-slice boundary, the relationship $$ G_R(x,x^\prime) = \Theta(x^0 - x^{\prime\,0}) K(x,x^\prime) $$ is general.
In quantum mechanics, the evolution operator $$ U(x,t; x^\prime, t^\prime) = \langle x | e^{-i (t-t^\prime) \hat{H}} | x^\prime \rangle $$ is a kernel. It solves the Schroedinger equation and equals $\delta(x - x^\prime)$ for $t = t^\prime$. People sometimes call it the propagator. It can also be written in path integral form.
Linear response and impulse response functions are Green functions.
These are all two-point correlation functions. "Two-point" because they're all functions of two points in space(time). In quantum field theory, statistical field theory, etc. one can also consider correlation functions with more field insertions/random variables. That's where the real work begins!