If you want to extend your stay in the United States, then you must file a request before your authorized stay expires. If you remain in the United States longer than authorized, you may be barred from returning and/or you may be removed (deported) from the United States.
Naturalization is the manner in which a person not born in the United States voluntarily becomes a U.S. citizen.
If you were born outside of the United States, there are two general ways to obtain U.S. citizenship through U.S. citizen parents. You can acquire U.S. citizenship at birth (acquisition) or you can derive citizenship from your parents after your birth but before the age of 18 (derivation).
A Permanent Resident Card (USCIS Form I-551) is proof of your permanent resident status in the United States. It also serves as a valid identification document and proof that you are eligible to live and work in the United States.