What a public inbox is, in plain language
A public inbox is a page that shows mail sent to one address, without asking who you are. If you can type the name, you can read the list. TheMailinator hosts that kind of page for addresses ending in @themailinator.com.
It is not an account
You do not register. You do not get a password. You do not get a private folder. The characters in front of the @ sign are the whole key. Short names and obvious names are the easiest for other people to open, so a name like admin or test is a poor place even for a harmless experiment you hoped to keep quiet.
It is not a sending service
This site does not give you an outbox. You cannot reply from the inbox page. If a test needs a reply, a thread, or a record you can search next month, use a normal mailbox you control.
It is not a records system
Messages can be delayed, rejected by the sender, or removed here later. The page shows a recent list, not a sworn archive. Do not use it as the only copy of something you need to keep, and do not use a screenshot of it as your only proof in a dispute.
A fair way to describe it
Call it a temporary, public receiving box for tests and other low-stakes checks. If a sentence about your plan sounds wrong once you add the word "public," choose a different tool. The path a message takes is described in what happens when a message arrives, and the cases to avoid are in when temporary email is the wrong tool.