While watching this video, at around 5:00, the man mentions a certain type of particle having imaginary mass. He also says that these kind of particles can go faster then light. But how it is possible that a mass is imaginary? For me, this make no sense. And also what does this even mean? For example, how would you calculate a gravitational force involving particles having imaginary masses? The result of the force will also be imaginary. Or will it interact gravitationally with other objects?
The idea at the most simplest level comes from the relation $E^2-p^2c^2=m^2c^4$. This is the Einstein relation $E=mc^2$ for a particle of mass $m$ traveling with relativistic momentum $p$. The idea is that if $m^2<0$, and therefore $m$ is imaginary, that $E^2-p^2c^2 < 0$ or $E^2<p^2c^2$. With real mass this relation is the other way around and corresponds to $v^2<c^2$, this results in the idea that imaginary $m$ yields $v^2>c^2$.
This isn't very physical, as you have noted with your confusion over implications of imaginary mass. It usually means that there is an instability in your theory, for example, in Quantum Field Theory, you are not expanding around the correct vacuum.
If you want to search for more discussion on the topic, we call these "particles" tachyons. See for example http://en.wikipedia.org/wiki/Tachyon.