One site I came across says Einstein's 10 Field Equations use 20 variables, while another said 40.
There are four variables in spacetime - three for space and one for time, right?
But there are two index symbols for each tensor, usually denoted by the Greek letters mu and nu. I don't know if the rank of a tensor is related to the number of variables in relativity.
A single vector equation like $\vec{F} = m \frac{d^2 \vec{r}}{dt^2}$ is actually three equations one for each of the $x,y,z$ components of the 3-vector $\vec{r}$.
In the spacetime of general relativity there are four directions to worry about, e.g., $t,x,y,z$. A vector is a rank-1 tensor and has four components. So a vector equation in GR would really be four equations. In common notation the components of the spacetime position 4-vector are $x^\mu$, where $\mu$ is an index designating which of the four components you are talking about.
In general relativity the gravitational field is represented by the metric tensor, which is a rank-2 tensor. In component notation it is written $g_{\mu\nu}$, where each index, $\mu$ and $\nu$, can take one of four values. This means it has 16 components. An equation, like the Einstein field equation, is actually 16 equations, one for each component of the metric.
The metric is a symmetric tensor, meaning $g_{\mu\nu}=g_{\nu\mu}$. This means some of the 16 equations are duplicates. Einstein's field equations are 10 independent equations that fully determine all 16 components of $g_{\mu\nu}$.
In general a rank-$N$ tensor has $d^N$ components, where $d$ is the dimensionality of the space, so $4$ for spacetime.
The Riemann curvature tensor is a rank-4 tensor, $R_{\alpha\beta\mu\nu}$, with $4^4 = 256$ components. Like the metric tensor, it has some internal symmetries (and anti-symmetries) so not all components are independent. It turns out there are only 20 independent components of the Riemann curvature tensor.
