Bathrooms are considered to be a wet area in your home, being highly exposed to water droplets and moisture. Each time you take a shower, all surfaces of your bathroom have to resist penetration from the resulting water and vapour.
Two areas in particular are most prone to water seepage. These are the grout joints between your bathroom tiles as well as the area where the wall and floor meet. At these points, gaps are filled by grout and sealed with caulk.
Over time, the adhesive in these gaps deteriorate due to the presence of excess moisture. Subsequently, your tile flooring becomes looser, enabling water to seep in slowly. If your tile loosens sufficiently, you could lift it up to check the sort of damage that has occurred below. You should expect to see that the lower floor has started to rot.
If your ceiling lies beneath a bathroom, then leakages from the ceiling may be due to the bathroom. In this case, as a temporary solution, you should make the effort to mop up water that is exposed to the bathroom. This would reduce the chances of water seepage and allow you to check if the ceiling leakages are indeed due to water from the bathroom.
With the level of exposure to water limited, you can now go about repairing your bathroom floor. At this point, you should engage a professional waterproofing company to inspect and repair your floor. Primarily, this would include determining the extend of damage to each tile and restoring them if possible.