In general, once a permanent resident has been in the U.S. in 'resident' status for 5 years (with trips abroad that add up to less than 1/2 the time during the 5-year period), s/he can apply to be a citizen of the U.S.
That's it, with exceptions, below.
There are exceptions for those who are eligible to apply for citizenship before the 5-year period of permanent residence, including:
1. Those who were born to U.S. citizens abroad;
2. Those who's parent naturalize to be U.S. citizens before they are 18 years of age;
3. Those who get residency through marriage to a U.S. citizen who are still with their U.S. citizen spouses at the end of 3 years (they can apply in 3 years rather than waiting 5 years);
4. Those who serve in the U.S. Armed Forces and their families;
5. And other select exceptions.
The government is very pro-active in publishing information regarding the naturalization process. Aggressive, even. Information related to the application (N-400) process is laid out at http://www.uscis.gov/us-citizenship/citizenship-through-naturalization. We are more than happy to answer individual questions @ email@example.com / 702.425.1770.
CAUTION . The naturalization process is the last time the government can review your eligibility in most cases. It means they look through the entire history of applicants to confirm all issues in the past have been addressed. These applications typically take 6-8 months, but they can take up to 5 years for background checks.
IF YOU HAVE ANY CRIMINAL CONVICTIONS IN YOUR PAST, PLEASE CONSULT AN ATTORNEY BEFORE APPLYING FOR NATURALIZATION.
IF YOU HAD ANY IRREGULARITY IN YOUR PERMANENT RESIDENT APPLICATION, PLEASE CONSULT AN ATTORNEY BEFORE APPLYING FOR NATURALIZATION.