A quick excerpt from Columbia Law School web-page explains:
“A Notice of Petition (also known as a dispossess) tells you what will happen if you do not come to court to answer your petition. A Petition tells you why your landlord is taking you to court. If you do not answer your petition promptly, the landlord will automatically win, and you will be evicted.”
Source: http://www2.law.columbia.edu/donnelly/hlas/Legal%20Aid/Tech%20Center_page_17_FAQs_NoticeofPetition.htm