Landlords may terminate a month-to-month tenancy simply by giving the proper amount of notice (30 days in most states). Reasons are usually not required. See the Nolo article How Month-to-Month Tenancies End for details. Leases expire on their own at the end of their term, and landlords generally aren't required to renew them.
If your tenant has done something wrong, you'll usually want the tenant out sooner. State laws allow landlords to do this by serving the tenant with one of three different types of termination notices, depending on the reason why you want the tenant to leave.
These are used when the tenant has not paid the rent. Pay rent or quit notices give the tenant a few days (three to five in most states) to pay the rent or move out ("quit"). See the state-by-state rent rules included on this site for the amount of time your state requires.
These are typically given after a tenant has violated a term or condition of the lease or rental agreement, such as:
Typically, the tenant has a set amount of time (this can vary from three to 30 days, depending on the state) in which to correct, or "cure," the violation—for example, by getting rid of the pet or the unauthorized roommate or toning down the late night noisy parties. A tenant who fails to cure the violation must move or face an eviction lawsuit.
A tenant who has violated the same lease clause two or more times (for example, by paying rent late) within a certain period of time may lose the right to a second chance. You may give the tenant an Unconditional Quit notice instead, which orders the tenant to vacate the premises within a short period of time, such as five to ten days, with no chance to pay the rent or correct the lease or rental agreement violation.
An Unconditional Quit notice may also be used for lease violations that cannot be corrected or cured because the effect of the violation is permanent. For instance, suppose your lease prohibits tenant alterations or improvements without your consent. If, without asking, your tenant removes and discards built-in bookcases, you cannot demand that the tenant cease violating the lease clause, because it is simply too late to save the bookcases.
If a tenant or guest substantially damages the premises or is engaged in illegal activity, such as drug dealing on or near the premises, you'll also be within your rights to use an Unconditional Quit notice. The law does not require you to give tenants accused of serious misbehavior a second chance.
If the tenant fails to leave in the time period specified in your Unconditional Notice to Quit, you may file an eviction lawsuit.
Many states have all three types of notices on the books. But, in some states, Unconditional Quit notices are the only notice statutes (in which case, landlords don't need to give tenants a second chance to pay the rent or correct the lease violation).
States typically have standards for the content (such as notice period) and look of a termination notice, requiring certain language and specifying size and appearance of type.
For the details and citations to your state's statutes on termination notices, see the Nolo charts State Laws on Termination for Violation of Lease and State Laws on Unconditional Quit Terminations. Also, see the Nolo article on Protections for Tenants Who Are Victims of Domestic Violence to see if your state law sets out some exceptions to your state termination rules.