The tenant eviction process varies depending on the type of eviction proceeding you're bringing, the court venue, and the length of the tenancy.
Generally you serve a tenant with the prerequisite notices required by law, then you draft, file and serve a petition with notice. You appear in court, and the judge will decide if your request for a judgment of possession and a warrant of eviction will be granted or not.
You can view a more detailed description of the process to evict a Long Island tenant here. And you can view a timeline of a sample evictions matter here.