Pool and vinyl liner manufacturers sell vinyl liner repair kits. These kits consist of a patch and adhesive. Sometimes the kit is provided with the pool or liner and other times, it's available as a separate accessory sold in pool supply stores.
Most vinyl repair kits specify applying the patch either after draining the pool or while the pool is still filled. It does not affect application to have the pool filled with water. Manufacturers of these types of patches produce them with adhesives that are water-resistant.
Cut the patch material into a round shape three times larger than the hole. Apply the adhesive to the back of the patch. Place the patch over the hole and apply pressure. Place a weighted object -- especially when applying it underwater -- on top of the patch until it completely adheres to the spot.