What should a landlord ask for if a tenant wants to use a cosigner to rent the apartment?
What information should the landlord ask for or need before he/she can approve the candidate?

OK, it is the landlord not the tenant that decides if there needs to be a co-signer.

The choice if the landlord wants a co-signer would be the credit history and the rental history of the tenant. If the tenant has had trouble paying their bills, been late on other rent or moved out and left a lot of damage before, you may want a co-signer but it has to be someone with money, a good job and someone you can actually sue and get something if the other person does not pay.

For a tenant , and also the co-signer if the tenant fails the test, you want to run a credit history, so a signed permission to run it and a non refundable amount to cover running the background check. You also want a complete list of all past rental over a number of years ( I ask for 10 years) and you personaly contact them for a background of their rental history.

And of course I get a full application, including work history, SS number, drivers license number, number of cars they will have and so on.