While cleaning is sufficient for day to day maintenance, you will come across other situations that calls for more drastic measures; e.g, cracked or loose tiles.
The most difficult step in replacing mosaic tiles is finding a matching tile. If you have leftovers from when you did the installation then you can skip this step. Try your best to find a matching tile. You can try repurchasing the tile from the vendor but be aware that there may be some slight variation in color and tone if the tiles are from different production runs. If that's fails, then you may want to try to replacing the damage tiles with a different tile to create an "accent". You should consider strategically picking and replacing neighboring tiles to further develop your "accent".
Now that you have your replacement tile on hand, it's time to remove the grout. Using a scoring tool, rake out the grout around the damaged tiles and any other you have chosen to replace. With the bad tiles separated from the rest of the design, you can proceed to removing these tiles by chiseling, prying, and scraping at it. Be sure to take your time and avoid damaging the neighboring tiles.
Afterwards you will want to remove the old thinset. You can verify that the thinset has been removed by placing your replacement tile into the void. If it sits lower than the other tiles, then you have successfully removed the thinset. Before you install the replacement tile, be sure to vacuum out any dust or remaining debris; otherwise the newly applied thinset may not bond properly.
To install the replacement tile all that is left is to apply the thinset onto the surface as well as the back of your tile. Then stick the tile into the hole and use a grout float to level the tile with the surrounding tiles. After allowing 24 hours of cure time, it's time to finish the job by grouting the tile. For more information on how to apply the grout, read 3. Grouting the joints.