As Marlow as already said, it is probably better (and definitely
more consistent with the terminology used by researchers) to treat
mass as an invariant and instead say simply that the energy of a
system depends upon its overall motion. So let's substitute the
word "energy" for mass, and "energy density" for "density".
To be more specific, let's take the system to be a cube of some
substance. The substance can be anything--wood, metal, or a box
full of gas or EM radiation. Now, as we put this system in motion
(or put ourselves in motion relative to it), there are two effects:
first, we measure its energy content to increase by a factor of
gamma, the usual Lorentz factor. Second, we measure its length,
in the direction of motion, to decrease by this same factor. Because
of the length contraction, the volume decreases by a factor of gamma,
and thus the energy density increases by a factor of gamma^2.
This is why energy density is the time-time component of a second-rank
tensor (the energy-momentum tensor). And *this* is one way of
understanding why the "field equation" for gravity is a tensor
equation (as opposed to electromagnetism, where the field equation
can be written as a four-vector equation, because the source, the
charge density, transforms with only one factor of gamma).