In order to become a U.S. citizen, most people first have to become a Lawful Permanent Resident, either through (link) Adjustment of Status or (link) Consular Processing. After having been a resident for the required number of years (three or five years depending on your personal circumstances), applicants can request Naturalization or U.S. Citizenship by filing Form N-400.
The citizenship process requires that applicants first submit their citizenship application, then attend an appointment where they provide fingerprints for a background check. Then, USCIS will schedule them for a citizenship interview at the USCIS office closest to where they live. At that interview, the individual will review their N-400 application and take an exam on their English fluency and U.S. civics/history knowledge. Some exemptions apply for English fluency, and some applicants qualify to take a simplified civics examination.
If the application is approved, the USCIS officer will schedule the applicant for their Oath Ceremony, which is a special celebration at which many new U.S. citizens are sworn in by taking the Oath of Allegiance to the U.S.