Naturalization is the process by which a foreign national becomes a citizen of the United States. Of course, there are certain requirements that must be met before one can become a U.S. citizen. Generally speaking, one must be 18 years or older, be a Lawful Permanent Resident (LPR) for at least five years preceding the date of filing the application, have continuously lived in the U.S. for at least five years prior to filing the application, pass an English and Civics exam, and pass a thorough background check. Among other things, USCIS will look for criminal records, any evidence of threats to national security, and any indication of bad moral character.
Having a criminal record can have a negative effect on your case. Under the Immigration and Naturalization Act, there are certain crimes that make certain lawful permanent residents, deportable. For example, a conviction for possession of a controlled substance, or even certain domestic violence statutes can mean that you will have to fight for your residence in Immigration Court. The crimes that do not make a resident deportable can still have a negative impact on a naturalization application. For example, if someone has a driving under the influence, an immigration officer can and may deny an application for naturalization.
To naturalize, you must have had five years of good moral character preceding the application. The idea of being a person of good moral character is very broad, and includes factors such as whether you have paid your taxes, and whether you have been arrested and convicted of a crime. Generally speaking, you should wait five years until after you are convicted of a crime. Of course, whether you should or should not apply sooner than that depends on the type of crime you have on your record, and even the sentence you received.
There are certain benefits to naturalizing, rather than remaining an LPR. Once a U.S. citizen, you can vote, petition other family members to come to the U.S. immediately, have your children born abroad derive citizenship, receive government assistance while traveling abroad, and qualify for federal jobs. Further, it is a way to show patriotism, and even shield yourself from any future removal/deportation risks. You should always speak to an attorney if you are interested in applying for naturalization and have a crime on your record.