Eggs
Eggs contain and are the main method of obtaining Pokémon. They require Egg Steps to hatch, and you can have up to six at a time.
Obtaining eggs
Adopt page
Eggs can be found all the time on the Adopt Page. Up to two eggs (or three with the Dowsing Machine Key_Item) can be seen at a time, and by reloading the page, you can see all the eggs available for adoption. Eggs on the Adopt Page do not have a name on them; they only have a description of what the egg looks like. The eggs tend to look like the Pokémon inside them.
Eggs adopted from the Adopt Page will have you as their original owner.
Pokémon Shelter
Eggs can also be found in the Pokémon Shelter. In the Shelter, you can filter by Pokémon, Eggs, or All. Eggs can be adopted from the Shelter without needing to spend a Shelter Ticket.
Eggs adopted from the Shelter will have a user other than yourself as their original owner.
Day Care Center
Pokémon that you already own can be bred in the Day Care Center to obtain eggs that are the same species as one of the parents.
Eggs bred in the Day Care Center will have you as their original owner when sent to the Shelter.
Event Eggs
Eggs can also be received as rewards for monthly Events, or from redeemable Event Tickets. These eggs also show up on the Adopt Page, but have a chance to show up on their own instead of next to another egg.
Event Eggs you redeem will have you as their original owner.
Hatching eggs
Eggs can be viewed and interacted with just like Pokémon. The only difference is that eggs don't have EXP; they have Egg Steps instead. Different eggs need different amounts of Egg Steps to hatch; in general, the rarer the egg, the more Egg Steps needed to hatch it. Once an egg hatches, it becomes a Pokémon and moves into your Active Pokémon storage, freeing the egg slot it was taking up and allowing you to adopt a new egg.
BIG IMPORTANT NOTE: Do not ask or provide any information about which eggs hatch into which Pokémon on this wiki or anywhere else. Per the Poliwager FAQ's rule, spoilers about eggs/egg descriptions are disallowed. As well, any other spoilers must be very clearly marked.