It is what it says it is, quantized flux.
The "typical" situation for this to occur would be superconducting ring, where you will find that the flux through the ring is quantized, meaning it is periodic, with phi_0 being h/2e.
This also gives rise to Abrikosov vortices (quantized "flux tubes") in type II superconductors. If you want to see some movies showing the movement of these you can just search for "vortex" and "superconductor" on Youtube.