In nonthermal plasma, not all particles move in the same way. The electrons are different from other particles. Both can be described as having a temperature separately. But that would mean, one piece of substance could have
Two temperatures at the same time
No temperature at all.
Both variants bend my mind just a little too much. It could be no substance at all, but that's what we call vacuum.
Is it just me, or is it one of the cases a human mind just can not bend around?
Thermodynamic equilibrium is not a trivial condition. Feynman, in his Statistical Mechanics lectures start writing that a system is in thermal equilibrium when all the "fast" things have happened and all the "slow" things not.
It may happen in some systems, for instance in the case of a plasma, that thermal equilibrium between electrons and between ions can separately be established, while processes allowing electron-ion equilibration may be much slower. In sucha a case one speaks about a two-temperature plasma.
Therefore, it is not the case of "no temperature" at all, but of a system with two sub-systems at different temperatures. Usually, the reason for having a long lasting difference of temperatures can be related to the presence of inefficient channels for transferring energy from the hotter component to the colder. Quite frequent situation, when there is a large mass difference.
The electrons are different from other particles. Both can be described as having a temperature separately. But that would mean, one piece of substance could have
Two temperatures at the same time
Temperature
First, I think you misunderstand what is meant by temperature in a system not in local thermodynamic equilibrium. Most of space is filled with a weakly collisional plasma, which means that the thing that usually keeps all particles in the air you breath at roughly one temperature does not mediate energy/momentum exchange in space. That is, Coulomb collisions are not creating an equipartition of energy.
In a nonequilibrium (e.g., see discussion at https://physics.stackexchange.com/a/177972/59023), kinetic gas like most plasmas, the "temperature" is more accurately described the by mean kinetic energy of the particles in a distribution in that distribution's bulk flow rest frame. This is mathematically illustrated at: https://physics.stackexchange.com/a/218643/59023.
Mulitple Temperatures
Because plasmas can be weakly collisional, it allows for multiple populations to exist with different temperatures (using the definition above). The low collisionality and huge difference in masses between electrons and ions changes the energy/momentum exchange rate or collision rate. Approximate expressions for the collision rates are given at: https://physics.stackexchange.com/a/268594/59023.
One can still describe a multi-species plasma with a single temperature (discussed at https://physics.stackexchange.com/a/375611/59023) given by: $$ T_{tot} = \frac{\sum_{s} \ n_{s} \ k_{B} \ T_{s}}{k_{B} \sum_{s} \ n_{s} } \tag{1} $$
What's more is that each population can have it's own communication speed, like a sound speed, in a plasma. This is because of the low collisionality and different masses of the different species which allows, for instance, electrons to stream past ions without colliding.
