If you only sublet from a private person, things can be very easy. It depends all on how much you trust each other. Often enough is
a copy of your ID / passport
a deposit
But if you rent directly from a landlord and become a main tenant (German: Hauptmieter), things are pretty formal. You will need:
Schufa-Auskunft a report of your creditworthiness by Germany's most known credit bureau
Mietschuldenfreiheitsbescheinigung: a confirmation that you have no debts from previous rents. You get this from your previous landlord
Proof of income (German: Einkommensnachweise): Usually from the last 3 months. And your income needs to be 3x higher than the rent incl. heating (German: Warmmiete)
Some apartments are part of social housing and require a WBS (German: 'Wohnberechtigungschein') from you.