You can easily replace your broken floor tile yourself without damaging any of the surrounding tiles. Follow these steps I followed right after hiring an expert St. Charles, MO basement finishing colleague to fix tiles in my basement; few weeks later, I incidentally broke one of them and I had no choice than to repair it myself to avoid further expenses. The process was very effective and easy to follow; 

1. The broken tile should be removed from the floor.

Use a chisel to remove the broken tile.  Put down drop sheets to protect all your surfaces from flying tile chips. Cover the centre of the broken tile with masking tape and then use a tile bit to drill in four or five holes. Use a chisel to chip away the middle of the tile and then wedge the chisel under the tile to gently pry it off, piece by piece

2. Clean the space where the broken tile was removed.

Use a grout removing tool to scrape the grout from the edges of the surrounding tiles. Chipping the old glue away with a paint scraper and sweep the area until you can see a dust free and solid surface.

3. Make sure the replacement tile fits properly

Place the replacement tile in place to ensure that it fits properly.

4. Install the new tile

Mix the glue you will be using to a toothpaste consistency. Cover the back of the new tile and then run a notched trowel across it. Place the new tile into the gap and press it down gently, make sure it sits evenly with the other tiles around it. Place masking tape across the tile and  wait for the glue to dry.

5. Grout the replacement tile

The next step is to remove the masking tape. If there are any excess glue in the gaps between the tiles, clean it up with a nylon scraper. Mix the grout, push it into the gaps between the tiles and smoothen it with a grout squeegee. For best results, run the squeegee across the gaps at a 45˚ angle.

6. Clean the excess grout off the tile

Leave the grout to dry for 20 minutes, then use a clean and damp sponge to wipe off the excess. If you are using the squeegee, you will get the best results if you run it across the tile at 45˚ angle. Rinse the sponge after every stroke to keep it clean and also squeeze it out so that it won’t be too wet. You can walk on the tile once the grout has cured for 24 hours.