It is easy for landlords and tenants to end a month-to-month tenancy in Texas. (The situation is more complicated when it comes to breaking a fixed-term lease.)
In most situations your landlord does not need to give you a reason (although acting on discriminatory or retaliatory motives is illegal). A landlord can simply give you a written notice to move, allowing you one month as required by Texas law and specifying the date on which your tenancy will end. However, the landlord and tenant may agree in writing to different notice periods, or none at all.
Your landlord may legally provide less notice in specific circumstances--for example, if you have not paid rent, if you have violated other terms of your rental agreement (for example, bringing in an unauthorized tenant), or if you have violated basic responsibilities imposed by law (such as by dealing drugs on the rental property).
It is equally easy for tenants in Texas to get out of a month-to-month rental agreement. You must provide the same amount of notice (one month) as the landlord. Be sure to check your rental agreement which may require that your notice to end the tenancy be given on the first of the month or on another specific date.
In some situations, you may be able to move out with less (or no) notice—for example, if your landlord seriously violates the rental agreement or fails to fulfill legal responsibilities affecting your health or safety.
Check Texas state law (Tex. Prop. Code Ann. § 91.001) for the exact rules and procedures for how landlords must prepare and serve termination notices and for any special rules regarding how tenants must provide notice. See the Laws and Legal Research section of Nolo for advice on finding and reading statutes and court decisions.
The state guide to tenants' rights (see Texas Attorney General, Tenant Rights) may also have useful information on how month-to-month tenancies end.