First of all, let me just say that in my personal opinion the nomenclature related to starlight is needlessly complicated and in need of trimming.
Anyway, stellar flux is the energy of radiation emitted by the star passing every second through a unit area. It is usually measured in a given band of EM radiation.
It's the same as radiative flux density or radiation flux if taken over all wavelengths and the same as irradiance if the area is a surface area.
If summed over a spherical area encompassing the source it gives the total power (luminosity) of the source. Luminosity over all bands is called bolometric luminosity.
Stellar flux features in the calculations for apparent magnitude, since what you're comparing there is how much energy arrives per unit time from stars at the surface of the Earth (i.e., how bright they look) versus reference flux.