Many lawful permanent residents want to eventually become U.S. citizens, i.e., naturalize. US citizenship brings with it a number of rights, including:
The ability to petition for the permanent residence of parents, children and spouse with no backlog
The ability to bestow citizenship on a child born abroad
A U.S. citizen cannot be put in removal proceedings
The right to vote, hold office, and serve on juries
The right to live abroad indefinitely

In order to be eligible to naturalize, a foreign national must meet all of the following requirements:

Must be Lawful Permanent Resident
Must be at least 18 years old
Must be a resident continuously for 5 years subsequent
to LPR status (3 years if the green card was based on marriage to a U.S. citizen
Must have resided for at least 3 months in the state in which the petition is filed
Must be physically present in the US for at least ½ of the 5 years
Must reside continuously within the US from the date the application is filed up to the time of admission to CITZ
Must not be absent from the US for a continuous period of more than one year during the period for which continuous residence is required
Must be a person of good moral character for 5 years prior to filing
Must be attached to the principles of the US Constitution
Must be willing to bear arms for the US
Must not be otherwise barred (subversive, terrorist, etc)
Must be able to satisfy the Civics and English language requirement