Bathrooms are often covered with ceramic tile, which poses a special challenge when hanging a new mirror. You can't nail the mirror in place because you will shatter the tile, but ordinary drill bits aren't up to the challenge of drilling through ceramic. Instead you need to use a special masonry bit that has been tempered for use with ceramic. Drilling holes into ceramic tiles takes time. If you are impatient and try to force the drill, you will have a mess to clean up.
Measure the space between the hanging clips on the back of the mirror.
Measure the same distance on the tile wall, and place a square of duct tape at each clip location.
Install a carbide-tipped masonry bit into a variable-speed drill.
Set the drill to its lowest speed. Too much speed simply causes the bit to spin against the ceramic, eventually getting red hot.
Drill a hole large enough for a plastic screw anchor in the centre of each duct tape square. The duct tape helps prevent the bit from wandering. Apply steady pressure, and cool the bit by dipping water onto it every 20 to 30 seconds.
Drill a hole at each clip location on the tile.
Push a plastic screw anchor into each hole. Most mirrors come with both hanging screws and anchors. If yours has only screws, purchase anchors to fit them at a hardware store.
Screw a hanging screw into each anchor. You should leave the screw extending slightly out of the wall to match the depth of the hanging clips on the mirror.
Hang the mirror on the screws, hooking the clips onto the extending screw heads.