How many days will the law allow a tenant to give up possession of an apartment if they are behind on rent and landlord wants them to move?

Your local landlord/tenant law will state the procedures for eviction and the amount of time from when notice is first given until the tenant must be out, before a court order can be obtained to physically remove them.

It could be anywhere from 14 days to several months, not including any delays for related hearings on counterclaims (such as fraud, health-code violations, etc).