Most of this stuff comes from <PERSON>'s book: 'The Meaning of Relativity'.
Riemann Curvature Tensor:
Imagine you take a curve, and you place a vector on one end of it. It doesn't matter which direction the vector is pointing. Now, say you displace that vector, to another point very close to it. Let's give them some names. The initial position of the vector is A and the final position is B. The vector itself is, say, $V^\mu$. lets say the difference between the coordinates of A and B is $\zeta$.
Now, this process of displacing a vector is known as parallel displacement. When $V^\mu$ reaches B, it has a change in it's direction ($\Delta V^\mu$). Measuring this change of direction gives us the curvature of the curve. It turns out to be $$\Delta V^\mu = -\frac{1}{2} R^{\mu}_{\sigma \alpha \beta} V^\sigma f^{\alpha \beta}$$ where $f^{\alpha \beta}$ depends on $\zeta$. Now, the bigger tensor above, is the Riemann Curvature Tensor. This tensor completely describes the curvature of the curve.
General Relativity is a theory which describes the curvature of spacetime, so the Riemann tensor has to be necessary in the mathematics of the theory. 418
The problem is: $$R^{\mu}{\sigma \alpha \beta} = -\frac{\partial \Gamma ^{\mu}{\sigma \alpha}}{\partial x_{\beta}} + \frac{\partial \Gamma ^{\mu}{\sigma \beta}}{\partial x{\alpha}} + \Gamma^{\mu}{\rho \alpha}\Gamma^{\rho}{\sigma \beta} - \Gamma^{\mu}{\rho\beta} \Gamma^{\beta}{\sigma\alpha}$$
Yes. It's just too tough to calculate. Each of the $\Gamma$s have yet more calculating to do. So it's a lot of work, with calculating each term takes you deeper down the rabbit hole; and so you could rather use:
Ricci Tensor:
This tensor describes the curvature of the curve, but is easier to handle than the <PERSON> tensor. So, how do you get it? First, let's talk about the Einstein Summation Convention. Basically, it is a shorthand for writing 'sum over indices'. If you have say the expression $$ds^2 = \sum_{\mu, \nu} g_{\mu \nu} dx^{\mu} dx^{\nu}$$ you can simplify and write $$ds^2 = g_{\mu \nu} dx^{\mu} dx^{\nu}$$ which means that if a particular index occurs on both the top and the bottom of an expression, you 'sum over it'. So, the <PERSON> tensor is $$R_{\mu \nu} = R^{\rho}_{\mu \rho \beta}$$ You sum over $\rho$ to get a more compact tensor. This is a process known as contraction Now, the <PERSON> tensor is the one actually used in the:
Einstein Field Equations:
Which are $$R_{\mu \nu} - \frac{1}{2}Rg_{\mu \nu} = 8\pi G T_{\mu \nu}$$ Here $R = g_{\mu \nu} R^{\mu \nu}$ (where the <PERSON> tensor is in the contra-variant form), and I have taken $c = 1$. Now, this is the central set of equations in General Relativity. The left hand side completely describes the curvature of spacetime, caused by the energy described in the right hand side. Or, as said by the oft-quoted line
Spacetime tells matter how to move; matter tells spacetime how to curve - <PERSON>
But the left hand side can be simplified. If we introduce the:
Einstein Tensor:
Defined as: $$G_{\mu \nu} := R_{\mu \nu} - \frac{1}{2}Rg_{\mu \nu}$$ Then the field equations become: $$G_{\mu \nu} = 8\pi G T_{\mu \nu}$$ that is another way of obtaining the Einstein Tensor $G_{\mu \nu}$.
In a nutshell:
The three tensors that you specified describe the curvature of space. 818