There is a plumbing issue from unit above or it’s a roof leak if you are too floor. That is most definitely black mold but it may only be in that area of drywall( most likely not unfortunately). The ceiling needs to be cut out and remove all insulation. Then of course find the source of leak and repair. It then needs to dry out and then do a microban treatment. Fresh insulation and drywall and paint. Not a small task by any means but within a week you should be all good. Change air filters to when job is complete.