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.
You must be careful. There are certain instances where applying for Naturalization can actually trigger the Bureau of Citizenship and Immigration Services to put you into deportation proceedings.