Your landlord can't evict you without terminating the tenancy first. This usually means giving you adequate written notice, in a specified way and form. If you don't move after proper notice (or reform your ways -- for example, by paying the rent or finding a new home for the dog), the landlord can file a lawsuit to evict you. (This type of lawsuit is sometimes called an unlawful detainer, or UD lawsuit.) In order to win, the landlord must prove that you did something wrong that justifies ending the tenancy.
State laws have very detailed requirements for landlords who want to end a tenancy. Each state has its own procedures as to how termination notices and eviction papers must be written and delivered to you ("served"). Landlords must follow state rules and procedures exactly.
Although terminology varies somewhat from state to state, there are basically three types of termination notices that you might receive if you have violated the rental agreement or lease in some way:
Even if you have not violated the rental agreement and have not been late paying rent, a landlord can usually ask you to move out at any time (assuming you don't have a fixed term lease) as long as the landlord gives you a longer notice period.
A 30-Day Notice to Vacate or a 60-Day Notice to Vacate to terminate a tenancy can be used in most states when the landlord does not have a reason to end the tenancy. (The length of the required notice can be slightly longer or shorter in some states.)
Rent Control Exceptions. Many rent control cities, however, go beyond state laws and require the landlord to prove a legally recognized reason for termination. These laws are known as "just cause eviction protection." (Tenants in only a couple of states -- New Jersey and New Hampshire -- also enjoy just cause eviction protection.)
Following receipt of a termination notice, if you haven't moved out or fixed the lease or rental agreement violation, the landlord must properly serve you with a summons and complaint for eviction in order to proceed with the eviction.
If you do get hauled into court, you may be able to diminish the landlord's chances of victory. Perhaps you can point to shoddy paperwork in the preparation of the eviction lawsuit. Or maybe the landlord's illegal behavior, such as not maintaining the rental property in habitable condition, will serve as a good defense, as would a claim that the eviction lawsuit is in retaliation for your insistence on needed, major repairs.
Even if the landlord wins the eviction lawsuit, the landlord can't just move you and your things out onto the sidewalk. Landlords must give the court judgment to a local law enforcement office, along with a fee. A sheriff or marshal gives you a notice that the officer will be back within a few days to escort you off the property. At that point, it's best to acknowledge defeat and leave on your own steam.
If you're a renter and want more information on evictions, see Every Tenant's Legal Guide, by Janet Portman and Marcia Stewart (Nolo).