It is common that you will have a ceramic tile with a crack in it, in a sea of perfectly unbroken tiles. If you are lucky you will have a replacement laying around you can remove the old tile and reinstall the new one. The trick is to get the old tile out without damaging the surrounding tiles.
Use a grout removing tool to remove the grout around the entire perimeter of the tile. There are a myriad of tools designed for this job, which include grout saws, carbide tipped tools, and a grout removing bit that works in a Dremel tool. Grout comes out quickly, just get down to the subfloor.
Now it’s time to remove the tile. As the tile is cracked, it may be loose enough to tap it out. You can use a chisel and a hammer and try to lift an edge of the tile. Be careful not to hit the surrounding tiles or they may break as well. You can also hit the tile with a hammer. Obviously, protect yourself with gloves, long sleeves, and protective glasses. The tile may shatter or you may dig a small hole with the business end of the hammer. In this case, use the hammer and chisel to remove the rest of the tile by leaning the chisel at approximately 45-degrees and striking the chisel until the tile pops out. Once the tile has been removed, clean up the old mortar by scraping the floor with a stiff blade. Then you can install the new tile.