From Bulbapedia, the community-driven Pokémon encyclopedia.
An Egg case is an anime-exclusive item, in which individual Pokémon Eggs are usually kept. Egg cases are small and compact, and act like incubators as they protect the Egg inside from harm. The glass container is cylindrical in shape; it has a yellow lid, a pillow-like blanket at the bottom on which the Egg sits, and a Poké Ball on the handle for putting the newly hatched Pokémon in.
Egg cases have been used for a number of Eggs owned by Ash and his friends. Ash's Phanpy and May's Eevee are two Pokémon to have hatched out of such cases. Manaphy had a different Egg case: the container was filled with water, had a gray top and bottom, and a light with a switch.
Egg cases are not used in the games. Instead, Eggs are carried in the party along with the rest of a Trainer's Pokémon. In Pokémon GO, however, there's an item called Egg Incubator, which the player can use to hatch Eggs. An Egg must be kept in an Incubator in order for it to hatch. There are two different kinds of Incubators: one which can be used an infinite amount of times and is in the player's inventory from the start of the game, and one which can be used to hatch three Eggs before it breaks and that can be bought with PokéCoins from the in-app shop, or given as a reward for gaining a level.
List of Pokémon that have been kept in Egg cases