A landlord can't proceed with an eviction lawsuit without giving notice (warning) to the tenant first. State laws set out detailed requirements as to how landlords must write and deliver ("serve") termination notices. If the tenant doesn't move (or reform -- for example, by paying the rent or finding a new home for the dog), the landlord can file a lawsuit to evict.
(For more information, see How Evictions Work: What Renters Need to Know.)
For all the legal and practical information you need to deal with your landlord, see Every Tenant's Legal Guide, by Janet Portman and Marcia Stewart (Nolo).