general relativity - What is the mathematical nature of the stress-momentum-energy tensor?

I am confused about the Einstein Field Equations. Specifically, consider

$$\begin{equation} \text{R}_{ab} - \frac{1}{2} \text{R} g_{ab} =\frac{8\pi G}{c^4}\text{ T}_{ab} \end{equation}$$

I understand $g_{ab}$ is the spacetime metric. I also understand that the left hand side of the EFEs gives us the curvature. But what is $\text{T}_{ab}$? I don't understand what kind of mathematical information it is giving me about the matter distribution or how this gives us info about the curvature of the metric.

It would be great if you could contrast a few examples of $\text{T}_{ab}$. I always see the example of dust or perfect fluids but I never understand why people think these examples are so useful or interesting. In fact, I had never heard of a perfect fluid until GR, so it has been a curious but fascinating way to be first introduced to them in this context.

REMARK: What is $\text{T}_{ab}$ called? I have heard the terms "stress", "momentum", and "energy" thrown around but I never know if all of them are used to describe $\text{T}_{ab}$ or just some of them.