Mix some matching paint (a paint store can match the color of your tile) with tile filler (Image 1), and dab it onto the crack with a cotton swab (Image 2). Allow the tile filler to dry for a few seconds, then wipe lightly with a damp cloth. When you're done, the crack will be much less visible (Image 3).
If the tile is too badly damaged to fix, you'll have to remove and replace it. Start by placing a rag over the damaged tile and then shattering it with a hammer (Image 1).
Remove the tile chips with a chisel, and clean out the old adhesive with a putty knife. Scrape around the edges of the remaining tiles to remove dust and debris that might get in the way of the replacement tile. Use a notched trowel to spread adhesive on the back of the replacement tile (Image 2), and set it in place.
Allow the adhesive to dry for the amount of time recommended by the manufacturer, then fill the grout lines with grout, available in squeeze tubes for minor repairs (Image 3). After the grout dries, wipe the tiles with a damp cloth or sponge.
If your grout has become chipped or stained, you may need to remove it. For that task you'll need a grout remover. After removing the old grout, install new grout and seal it to minimize future stains and damage.