Some Lawful Permanent Residents don't try to become US citizens. Perhaps they don't want to. Perhaps they do want to, but they don't think their English is good enough or they don't think that they can pass the test. Becoming a US citizen is called naturalization, and is as easy, and as natural, as falling off a log. However, it is almost always the case that before becoming a US citizen a person has to be a Lawful Permanent Resident first. As a US citizen, you have certain benefits. For example, as a US citizen, you gain the right to vote. You also don't need to worry about being deported simply for having committed some crime. (Unless you are encouraging terrorism. Be careful of the charities you donate to.) Speaking of crimes, if you have committed a crime, make sure that you are not throwing yourself into deportation proceedings by filing an application for naturalization. Many crimes can be modified by an experienced criminal attorney to make them 'immigration friendly' or 'naturalization neutral.' Then again, some crimes won't have any impact on your naturalization application, modified or not. The only way to know for sure is to contact a knowledgeable immigration attorney.
Not everyone can become a US citizen. Only certain people are eligible. In most cases, to be able to submit the form N-400, the application for naturalization:
1) the applicant has to be at least 18 years old;
2) in most cases the applicant has to have been a permanent resident for at least four years, nine months (unless the applicant has been married to the same US citizen for the past two years, nine months);
3) the applicant has to be a person of good moral character;
4) the applicant has to have a basic knowledge of United States government and history;
5) the applicant must have been physically present in the US for a certain amount of time; and
6) the applicant must be able to speak, read and write basic English. This last requirement is scary for a lot of people. Actually, the writing part of the test is easier than one might think, if the applicant practices writing simple sentences with the naturalization vocabulary the government has made available. Not everyone needs to be speak, read, and write basic English. If you are at least 55 years old and you've been a Lawful Permanent Resident ("LPR") for at least 15 years, then you can take the exam in your native language. If you are at least 50 years old, and have been an LPR for at least 20 years, then you can take the exam in your native language. Also, an exception is made for certain people who are physically or mentally impaired, and who are unable to meet the above requirements because of this impairment.
This article hardly touches the surface of the naturalization process. For more information, please visit www.uscis.gov and get the M-476 Guide to Naturalization.
Disclaimer: This comes as general information, and is not intended for specific applications. I suggest discussing your personal details with an experienced immigration attorney.