Heat the tile up by using a clothes iron. This helps loosen the remaining adhesive so the tile will pop out more easily.
Pull the tile up with a pry bar. If you can't get it out easily, you can use a hammer and a chisel to chip the tile out, but if it is still affixed firmly enough that this is necessary, you should maybe not replace the tile, and simply help fasten it back in place.
Remove the mortar or adhesive from the floor and from the back of the tile if you plan to reuse it instead of replacing it with a new one. Scrub the surfaces with a wire brush to lift up mortar, or use a disc sander to quickly take it off. Be sure to wear a mask if using a disc sander. Clean up the area thoroughly so you have a clean subfloor to install the new tile on.
Mix the thin-set mortar according to the directions on the label. Trowel a level bed of the mortar into the opening you have created, and then set the tile firmly in place. Gently strike the tile with a mallet to ensure that it is settled firmly in the bed. Wipe away any excess thin set that pushes out and gets on the surrounding tiles.
Allow for the thin-set to dry completely according to the package instructions without walking in or disturbing the tile in any way. Once that is complete, you can put grout that matches that original grout job around the tile to finish the replacement.