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 the protagonists. 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.
List of Pokémon that have been kept in Egg cases