To get rid of a petz, you can sell it in the shop (only if you pixed or bred it) or trade it away.
If you are absolutely certain you simply want to abandon it, click the pencil icon in the top right corner of its profile and then click the trash can button. This will send it to the Pound (you can search for the Pound the same way you search for other players, but they are not a real player).
Pregnant and newborn petz can not be abandoned.
By default, everyone has 8 boxes to sort their petz. On a single-time purchase of the Welcome Bundle, you can have 50 boxes forever. The number of boxes is only for sorting your petz, and is unrelated to box space.
The user "Pound" is not a real person! It's a holding account for abandoned petz.
Unfortunately, you cannot currently adopt petz from the Pound unless you are the pixer, but sometimes if you notify the pixer they can adopt it and sell it to you!