ANDREW & HILARY MEREDITH
First you need to get it up without damaging the ones next to it. if it is badly damaged that will be easy - put something like a screwdriver into the broken part and lever it up from there. If it is only cracked you need to carefully dig out the grout between it and the neighbouring tiles - again, a flathead screwdriver can be used - and then get under the tile and lever it up, taking care not to damage the edges of the tiles that are ok (it might help to put cardboard under the edge of whatever you are using to lever it up). When that is done, see how much dry tile adhesive is left underneath and scrape most of it away. Once more, the screwdriver comes in, possibly helped by a hammer. The main thing is to get rid of anything that would stop the new tile from laying flat. When that is done and all the bits removed (pick up lumps and vac the rest up - very important to get rid of dust because it will prevent the new tile from sticking down properly), put some tile adhesive on the back of the new tile (always assuming you have one!), wiggle it about to get it level (if necessary get it up again and put more or less adhesive on it), and leave to dry for a day. Then fill the gaps with grout and smooth off. You can sponge off excess, but to get the final finish on the new tile, wipe with a dry cloth when the surface has dried - a final dry polish, in other words. Normally you would use floor tile adhesive. If the floor is solid (such as concrete), then you can get away with wall tile adhesive for one tile. Just don't tread on it till next day! (The special stuff for floor tiles sets more quickly, but you don't want to have to buy a whole bag, whereas you can get ready-mixed wall tile adhesive-and -grout -in-one in a small container form a DIY store.) But if you have coloured tiles with coloured grout you will have to get some of the appropriate colour, possibly from a specialist tile shop. That will be wasteful but you don't (one assumes) want a bit of the floor standing out in the wrong colour. Finally: the reason why you don't grout immediately is to allow the air to get under the tile so the adhesive can dry hard. Even if you use the all-in-one stuff, i.e. the same thing for both purposes, you still need to wait. Always check what it says in the product instructions.
