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You referred to rest mass in your question. This is generally what physicists talk about when they say "mass" in the context of modern physics. Rest mass is quite simply the mass something has in its own reference frame; its mass when it is not moving. Photons have zero rest mass. Anything with zero rest mass always moves at a speed c, which we call the speed of light.
Electrons, protons, neutrons, and pretty much everything else that we deal with in day to day life have rest mass. Anything with rest mass naturally gains more and more energy as we speed it up. This energy is equivalent to "inertial mass". Inertial mass is what you multiply the velocity by to get the momentum. One could think of this as the object gaining more "mass" as it speeds up, but physicists usually don't use the term that way.
The idea of "inertial mass" has an intuitive appeal in special relativity though: as you speed something up more and more (giving it more energy) you also increase its inertial mass. Earlier I said that inertia is "how hard something is to push". If the object is gaining inertia as it speeds up, then it would imply that it also gets harder to push. Well that is exactly what happens. When you take something and try to accelerate it to a significant fraction of the speed of light, it gets increasingly difficult to make it go any faster. This is why particle accelerators like the Large Hadron Collider are so complicated. They accelerate particles to 99.9999991% the speed of light, which requires monumental amounts of energy.
Another potentially appealing property of relativistic mass is that it is additive. This makes the quantities easy to deal with conceptually, since they add just like a classical mass would. The inertial mass also is the quantity that enters into gravity, so light does indeed have weight, but not much.
In conclusion, Einstein wasn't wrong, but the terminology needed to understand special relativity can get confusing and is often misinterpreted.
Matt J.
(published on 02/28/2012)