Aren't all objects luminous in a sense? In my physics class, I learned about "nonluminous objects" - these are objects which don't produce their own light. But, don't all objects emit light by black body radiation? So are all objects luminous objects? (except objects at 0K, if we get there somehow), or am I missing some point here?
 A: Yes, all objects emit radiation.  But without other considerations, I would interpret "nonluminous" to simply mean that it isn't emitting visible light.
My cat is quite warm, but I would use "nonluminous" to describe it unless we were specifically discussing light as any form of EM radiation.
A: As you say, all matter is radiating, even black holes! In principle, the phrase "nonluminous objects" should be qualified by what part of the spectrum they can be considered nonluminous and how much flux is needed to consider it "luminous". When people say "nonluminuous" they probably usually mean something like: the object emits less than 1 µW/m2 on the 400–700 nm range of wavelengths (in air or vacuum).
A: Black bodies are in equilibrium with their surroundings - they absorb radiation from their surroundings and then re-emit it. Luminous bodies have internal energy sources, i.e., there is energy produced within these bodies, which is then emitted in the form of radiation. In this sense these bodies are not in thermal equilibrium, as there is a constant energy transfer from within the body to its surroundings, and this energy does not return back to the body.
Thermonuclear reactions within the Sun are an obvious example. Note however, that the radiation emitted in this way is reabsorbed/scattered many times before it reaches the surface, which results in the radiation spectrum being very similar to the black body spectrum - if the Sun is considered as isolated, the radiation reaching its surface can be considered as thermal/equilubrium radiation. However, the Sun is obviously not in thermal equilibrium with the surrounding vacuum (they have different temperatures - 6000K vs 0K).
See also How does radiation become black-body radiation?
Remarks:
@Quillo has pointed out in the comments that the energy is not necessarily generated within a luminous body, but could be simply stored in it (aka fossil heat) and being gradually released to the environment in a form of radiation. The body is still not in equilibrium with the surroundings, as it releases more energy than it absorbs.
@JonathanJeffrey has pointed out that the correct value for vacuum temperature is 3K, due to the microwave background radiation.
