Hi MsTee - Yes, you need to look at the lease the person signed, unless they did not sign a lease. I am not sure specifically about New York, but often if there is no lease, it means it is a month-to-month contract. If that is the case, you could probably give them notice 30 days out.
But as kprimeaux said, if they have a signed lease, you can't just kick them out without them violating the lease in some way. And even then, evicting a resident is quite hard. Often, judges tend to be very pro-tenant, so you must follow all guidelines to the letter when trying to evict.
Hope that helps!