A permanent resident of the United States can remain in the U.S. for an indefinite period. There is no requirement that a permanent resident become a citizen of the U.S., but a permanent resident may choose to do so. To be eligible to apply to become a naturalized citizen of the U.S., a permanent resident must have lived in the United States in that status for 5 years. A shorter 3 year time applies to persons who have lived in the U.S. as a permanent resident spouse of a U.S. citizen.
Forms and information for applying for naturalization are available from the INS webpage at http://www.uscis.gov/graphics/services/natz/index.htm . Processing times for naturalization applications vary across the U.S., but processing commonly takes 2 or more years.