Flux refers to flow.
Flux density (or, commonly, flux for short) is the rate of flow of something per unit area, in fluid dynamics, magnetism, electromagnetic radiation and other fields.
Flux can also refer to total "flow" from something, such as EMR from a star, in the phrase radiant flux. It is also used, e.g., for observation with CCDs, to refer to the total signal received by a region of it, such as an individual pixel.
Fluence (aka radiant exposure, in the case of EMR) is a total amount of flow per unit area rather than a rate. For example, the fluence of a gamma-ray burst indicates the total energy it put on each cm2 over the course of the entire burst (whereas the burst's flux density at some point in time would indicate the rate at which it was putting energy on each cm2 at that point in time).