Incenses will be used for the upcoming breeding feature. Certain incenses also give benefits to the holder while in battle. They can be crafted or dropped by boss Pokémon. Some of them (Lax Incense, Odd Incense, Pure Incense, Rock Incense, Rose Incense, Sea Incense, Wave Incense) can be found in PokéLoot Chests.
Gems are single-use held items that boost the damage of a move of their corresponding type by 1.5x. They can be crafted. The Fairy Gem is an exception that does nothing and cannot be crafted.
Mail is a special type of held item that is used to write messages. It can only be obtained by spawning it in through creative mode or other means.