If a tile wobbles, it means that the adhesive behind has failed and the only thing holding it in place are the grout lines. Once the grout has been removed you should be able to lift the tiles out of the floor.
If you have just one or two tiles that are loose, you can use a grout saw to cut through the grout. If it's just the grout that's holding them in place, they should be easy to remove.
If it's a more extensive job with several tiles, you might like to invest in a multi-function tool and a diamond attachment for grout removal to remove the tiles.
If the tile does not wobble before it is removed, then it is likely that the adhesive is still stuck to the tile. Unfortunately, there is no easy way to remove the tile without damaging it. If the tile needs to be removed it will probably need to be broken with a mallet and a new tile will need to be installed. It is strongly recommended that you have spare matching tiles before you begin your project.
In order to glue the tiles back in position, you'll likely have to remove some of the existing glue from the back of the tile or floor. If you just glue them back down without removing the existing glue, the tiles will sit slightly proud of the surrounding tiles.
Here are a couple of helpful step-by-step guides: How to remove grout and How to lay floor tiles.
Please remember to wear personal protection such as gloves, goggles and a mask when working on your project. - MitchellMc
