Thermodynamically pressure is related to the work done in changing the volume. Assuming an adiabatic change the change in the internal energy of a system is given by the work done on it:
$$ dU = \delta W = PdV $$
If you take a volume of an ideal gas and let it expand by $dV$ it does work on it's surrounds and it's internal energy decreases. This is the usual (positive) pressure.
But consider the dark energy that we believe pervades the universe. If we take some volume of vacuum and let it expand by $dV$ then the internal energy goes up. This may seem odd, but it goes up because dark energy density is a constant everywhere so a bigger volume has more dark energy. Anyhow, if the internal energy of our system goes up when the volume increases we must have done work on it i.e. the sign of $PdV$ is negative. Since the sign of $dV$ is positive, because the volume increases, we conclude that the sign of $P$ must be negative i.e. dark energy has a negative pressure.
Dark energy is not equivalent to negative mass in the $E = mc^2$ sense. The energy density is positive. The reason it shows a negative pressure is due to it's equation of state and not to it being in some sense negative energy.