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What accounts for the variation in the universe,

Believe it or not, acoustics. I'm sure you've heard that there's no sound in space, but that's because the universe is so rarefied. That didn't used to be the case; sound waves were the main energy transport in the early universe. The result is that you get a characteristic spectrum of density fluctuations. You can see in the following plot just how well the observed fluctuations match the theoretical acoustic spectrum:

and what accounts for the fact that the universe is mostly empty?

Gravitational collapse. The denser areas collapsed into stars, galaxies (and even larger structures), which sucked the matter out of the less dense areas.

I've run a few simulations of such cosmological structure formation using the National Scalable Cluster Project supercomputers here at Penn, using parallel code written by Prof. Paul Bode.

This animated .gif shows the mass density of a simulated portion of the universe 100 light-years across. Black is lowest density, followed by blue, green yellow, red, and finally white as highest density. Later images show later times, when more structure forms due to gravitational attraction. As the movie plays, the universe ages from 50 million to 12.5 billion years. During this time, our simulation volume actually expands by a factor of 40, but this expansion isn't shown.
Here is another such universe, showing more resolution (WARNING: 7 MB).
Here is the first frame of that universe, and here is the last frame.

It's an interesting research topic.

I understand your answer, and thank you for same, but I respectfully disagree. To me, it is difficult a transition from a singularity to a non-singularity that is not completely uniform, UNLESS there is manipulation of the system, externally or internally. Implicit in my skepticism, I suppose, is the question begged: What would CAUSE a singularity to change state, and whatever the explanation could be, HOW would it, as the "first cause," influence the transition? This is not intended to suggest a deliberate "design," but rather simply the need for some "force" to cause the transition from a singularity to be something other than uniform. I still cannot fathom how a singularity that somehow "explodes" does so in anything other than a completely uniform, concentric "shape." If, at some later point in time, the matter is affected by gravity, effect of sound waves, or other "shaping" influences, how can it transform from anything other than concentric in shape? Unless there are other unaccounted for forces affecting the post-Big Bang explosion that are themselves not uniform in existence. I appreciate your answer, though. Do you have any comments about my comments? This is fascinating stuff, isn't it?