There are many different types of lease agreements in New York. There are some things that all leases cannot legally do and there are certain tenant protections regardless of what a lease states. However, leases vary in length, renewal process, amount of rent, termination process and other terms governing the landlord/tenant relationship and potential landlord-tenant disputes. If it is a rent stabilized housing unit, there must be a current lease, but this is not true if the building is not rent stabilized.
When a renter has a lease it is easy to determine when rent is due, how long the tenant can live there and other basic terms. However, if a renter does not have a lease this can be more difficult. So, by default if a renter pays rent each month, which is accepted by the landlord, there are considered month-to-month tenants. Also, if it is not a rent controlled building, if a tenant stays past the end of the lease and the landlord continues to accept rent they are considered month-month tenants. This means that the rental agreement lasts one month and renews when they pay rent the next month.
In order to terminate a month-to-month tenancy, either party must give 30 days notice in New York City or a month's notice outside of New York City. Neither party needs to give an explanation for why they are terminating the lease. Also, rent may be raised without the consent of the tenant. If the tenant does not agree to the rent increase, the landlord can terminate the lease with proper notice.
It is best to have a current lease in New York to protect the tenant's rights. However, if the tenant does not have a current lease, they still do have some general rights and are considered month-to-month renters. It is important for tenants to understand their rights though whether they have a lease or not and experienced attorneys may be a good resource.
Source: ag.ny.gov, "Tenants' Rights Guide" accessed on Aug. 24, 2017