If you've got a cracked or missing roof slate rain water will get in and could cause considerable damage. You'll need to replace the broken slate. Sometimes you may find the slate is not damaged and has simply slid out of place (probably caused by corrosion on the securing nail)

The tricky thing is due to the way slates are laid on a tile roof. They are nailed onto the roof battens starting from the bottom. The next tile up is overlapped thus covering the nails. So the problem when removing the odd tile is to get it out and secure a new one in place without disturbing the surrounding tiles

With the old tile out the new one will simply slide in but you need a mean of securing it. Several method are available.

For a quick temporary fix you could simply glue the new one on place with some high performance instant grab adhesive like Gripfill. Make sure that under the tiles are clean and dry. And use a weatherproof waterproof adhesive.
Remember, you'll need an applicator gun for this product. (see my tools section)

For a more permanent fix:
Make yourself a "tingle" from a piece of sheet lead or aluminium, or any non rusting metal that is easy to bend.

Cut a long strip approx. 25mm. wide and drill a small hole in it. This is nailed in the gap between the underlying tiles onto the wooden roof batten.(you should be able to estimate where the batten is underneath the roofing membrane by looking at the position of the tiles and comparing where the nail hole is on the new tile). Make sure you use a galvanised (non rusting) nail.

With the "tingle" nailed in place slide the new tile in place on top of it. You need to plan the length of it so that a small amount is left sticking out. Finally bend the sticking out bit over the tile to form a hook which stops the tile slipping down.

NB WARNING Working at heights can be dangerous. You must ensure that you use appropriate access equipment. It is your responsibility to ensure your own safety.