How does rotation of a mass affect the geometry of space? I understand (or rather, I have been told, but do not understand) that a large mass can distort space by its rotation. While I am somewhat familiar with the concept of a mass distorting space and thereby creating what we experience as gravity, I haven't been able to find any explanation of how the rotation of such a mass can be a factor and how it would distort space differently from a similar mass at rest.
Can anyone enlighten me?
 A: Actually, the explanation as to why rotation of a mass affects the metric in principle is simple. Rotation means there is angular momentum, and angular momentum contributes to the energy-momentum-stress tensor in general relativity. If this was a nonrelativistic rotation we would say that the rotation carries kinetic energy. 
The rotation contributes as a source of the gravitational field from a rotating mass. That's because rotation carries energy (in a sense). The curvature, which is a more technical terms for what you call the distortion of space, and in general relativity where the metric depends on the coordinate system used one must use invariant (scalar, meaning independent of the coordinate system) measures to describe it, is higher than that of a non-rotating mass. The Kerr solution is the external metric for a uniformly and constantly rotating mass, and is thus the metric of the spacetime outside the body. The curvature invariant are known and can be calculated.  
The Kerr metric is also the metric of a rotating black hole in equilibrium, and it is known that the rotation, or more exactly the angular momentum, carries energy. For instance it is possible to extract energy from a Kerr black hole at the expense of a lower angular momentum, and have that energy go to the kinetic energy of matter outside the black hole (this is the Penrose process). 
The angular momentum of a massive body also causes the coordinate frames outside the rotating body to be dragged along the direction of rotation. 
The exact effect of angular momentum or rotation, for constantly rotating bodies (in general relativity the technical terms are: for axially symmetric stationary spacetimes, in vacuum - meaning outside the body) is well known from the Kerr metric. The equations of the resulting metric and calculations of effects have been done, and you don't need to know everything to understand at least some of it. The Wikipedia article for the Kerr metric is more sophisticated than it needs to be. If you want to understand more Google Kerr solution, maybe find easier explanations, or somebody else can recommend something. Another way is google the Susskind lectures on general relativity, he explains things physically with simpler math.
Remember that in Relativity energy and mass are in some way equivalent by E= mc^2. In General Relativity all forms of mass and energy, and in fact momentum and stresses, contribute to cause a gravitational field. The rotational energy is equivalent to a mass by that equation, and contributes to the gravity. 
A: The bottom line is that space and mass do interact with one another. Otherwise, they would not tell one another "how to curve" and "how to move".
Therefore a moving (or rotating) mass and space will interact slightly differently. The interaction will drag with rotating mass. It may or may not be detectable depending upon the mass & speed of rotation and the sensitivity of the detecting equipment.
