Is it mass or density that curves space time?

On the left side of Einstein’s field equations for General Relativity is the Einstein curvature tensor $G^{munu}$. On the right side is the energy-momentum-stress tensor $T^{munu}$ of all non-gravitational “stuff”: matter, radiation, Higgs field, etc. The 16 components of $T^{munu}$ represent the density and flow (or “flux”, in 3 directions) of energy and momentum.

Einstein’s equations state that the two tensors are simply proportional:

$$G^{munu}=frac{8pi G}{c^4}T^{munu}.$$

(The constant of proportionality involves Newton’s gravitational constant $G$ and the speed of light $c$, as one would expect for a theory of relativistic gravity.)

So, based on the relationship between the two tensors, it is common to say that spacetime curvature is “caused by” the density and flow of energy and momentum. (Note: Another word for “flow of momentum” is “pressure”. Pressure gravitates!)

Suppose we are talking about a collection of atoms and we want to know how they gravitate. In addition to knowing their masses and their locations, as in Newtonian gravity, in General Relativity we also need to take into account their velocities, because this motion affects the energy flow, the momentum density, and the momentum flow. Some people call these gravity-of-motion effects gravitomagnetism.

Talking about just mass or “density” (density of what?) curving spacetime is common in popular introductions to General Relativity, but it’s not the whole story. $T^{munu}$ is the whole story about what “causes” curvature. So the next step in learning about this would be to read the Wikipedia article I linked to, which goes into detail about this tensor.

Note: I am taking the liberty of not bothering to distinguish between the components of a tensor and the tensor itself. Mathematicians hate this, but physicists do it frequently; entire textbooks on General Relativity, such as Weinberg’s, have been written this way. Today, it is somewhat old-fashioned as physicists increasingly adopt the mathematicians’ more abstract and often component-free notations, which do a better job of conveying the geometric nature of the physical quantities. I have also ignored the question of a cosmological constant term, and whether to put it on the left side with the curvature or incorporate it into $T^{munu}$ as “dark energy”.