The Shelter is a place for PF (PokéFarm) users to claim abandoned eggs as their own. Unclaimed Daycare eggs are sent here as well as Pokémon. Pokémon often move around in the shelter. To stop them from moving, just roll your cursor over the Pokémon. At the bottom, there should be a box with the information about that egg/Pokémon that you have your cursor over.
Th Shelter

The Shelter

When you click a Pokémon/Egg, you will have the option to put it back in the Shelter if you don't want it. If you adopt a Pokémon, you will have the option of naming it. Once you have claimed that Pokémon/Egg, you cannot release it for the next 12 hours, so be sure to make up your mind before claiming it. If a Pokémon is not claimed after a certain amount of time in the shelter it will be released to the wild.

Keep in mind that in order to adopt a Pokémon from the Shelter, you must first have a Shelter Pass. You obtain the Shelter Pass as a reward from hatching your first egg.