Zahava from Toronto, Ontario, has a tile that has come loose in a new floor and it makes a gritty sound when you walk on it.
Start by cutting away all the grout around the offending tile. Cutting away the grout will liberate it from its neighbours. Then lift it out. It won't be stuck down because if it was, it would not make that gritty sound.
To remove that grout you can use a little hand tool that you see in the first photo which is specifically made for cutting grout -- a carbide dusted blade with a little plastic handle. For power tools there are special grout removal attachments for all the rotary tools, like Dremel or now commonly available are Oscillating Tools that do a great job if grout removial. There is a video on this in the Learning Curve Tab above on Digging Out Material, or just click here.
If you are trying to remove a broken tile that is stuck down, remove all the grout as before, then drill a hole and work from the centre out. This will avoid bothering the neighbouring tiles.
Clean off the mortar from both the floor and the tile using a Carbide rasp - the FEIN MultiMaster does a great job of this as it has specific blades for specific jobs. You could use the same thin set mortar used to install the original tile but you don't really want to buy a whole bag of this stuff for one tile. For a single tile installation you can use a tube of PL Premium construction adhesive. The reason I choose PL Premium for this is that this particular adhesive will eventually set rock hard rather than remaining flexible as many other adhesives do, and it doesn't shrink. Put in enough to fully support the tile and hold it up to the right height. If your excess oozes out into the grout area, scrape it out of the crack immediately and don't spill it on the top of the tiles, it may stain the tiles and it will stain your skin, although it is non toxic. Give it several days to set hard. Then re-grout around the tile. This one will never move again.