Naturalization is the process whereby people who have had their Green Cards for a certain period of time can apply to become United States Citizens. Becoming a United States Citizen gives you many privileges that are not afforded to a Lawful Permanent Resident.
Generally, if you have been a Lawful Permanent Resident for five years, you are eligible to apply. Additionally, if you have been a Lawful Permanent Resident for three years and are married to a United States Citizen, you are eligible to apply for United States Citizenship in only three years. The petition itself can actually be submitted 3 months prior to the time where yo become eligible.