Notice to quit
A written notice from a landlord that ends the tenancy and tells the tenant to leave. In New Jersey it does not force you out on its own; a court must enter a judgment for possession before you have to move.
New Jersey example
Say a Jersey City tenant keeps breaking reasonable building rules. The landlord first sends a notice to cease. If it keeps happening, the landlord must serve a notice to quit at least one month before filing for eviction. Other causes have other notice periods, like three days for disorderly conduct. An eviction for nonpayment of rent needs no notice to quit.
Official source
This is a plain-English summary. Rules and dollar amounts change, so check the official source or a professional before you file or sign anything.