The following photographs and text deal with a broken soap dish. However, the principles for the soap dish replacement apply also to other type of fixtures and individual tiles.
The first step in this process is identifying the problem. In the case of this soap dish the problem is obvious. In the case of loose tile or "hollow" sounding tile, there may be another alternative to replacement. Look in the section "Repair Loose Tiles" for more on that subject.
For this job a total replacement was necessary. The first thing to do was to obtain a replacement for the broken soap dish. Measure the old one and buy a new one that will fit properly.
Properly mask off adjacent surfaces so that they will not be damaged in the removal step.
Cut away the existing grout or sealant from the perimeter of the fixture. This will help prevent the chipping or damage to adjacent tile.
Next use a combination of chisel and hammer to remove the old fixture.
Be sure to wear safety glasses. There will be flying chips of very sharp ceramic.
Once the balance of the fixture is removed, be sure to scrape down the area below the dish to ensure a snug fit.
Vacuum out the area or brush out the dust left from the removal process.
Mix a small amount of thin set mortar according to the Manufacturers instructions. Back butter the dish and "key in" the thin set to the soap dish space. This will ensure a good bond.
Place the soap dish into position and align making sure to cover the original hole as close as possible. Completely dry the fixture and hold it in place with strips of duct tape until the thin set can cure.
The final step is to use a sealant caulking on the perimeter joint and the job is finished.