A one-page form, sometimes called a Wohnungsgeberbescheinigung, on which the person providing your housing states their name and address, the address you are moving into, and the date you moved in. They sign it. You take it to the Bürgeramt. That is the whole document.
It exists because Germany abolished a system where you could register anywhere you liked and replaced it with one where a real person has to vouch that you live where you say. The obligation runs the other way too: your Wohnungsgeber is legally required to give you one within two weeks of the move, and is not supposed to charge for it.
In a sublet, the Wohnungsgeber is the main tenant, not the owner. This trips people up constantly — you do not need the landlord's signature, and you do not need to be on the original lease. What you need is the signature of the person who is actually housing you, which is the person you are paying.
Where it goes wrong is a host who does not want to sign, usually because their own subletting is unpermitted. There is no clever workaround: without the form you cannot register, and the Bürgeramt will not take your word for it. Which is why this belongs in your first message rather than your fourth week — “would you be able to sign a Wohnungsgeberbestätigung?” is a fair, ordinary question, and a host who says no has just saved you a great deal of trouble.