It depends on what kind of visa you have. You cannot become a permanent resident based on a tourist visa or most other non-immigrant visas. In general, you can become a permanent resident based on a family member or employer’s sponsorship or your status as a refugee.
Adjustment of status is the legal step of applying for permanent residency (also known as a green card) in the U.S. and changing your legal status in the U.S. from a visitor to a legal permanent resident. This is done via Form I-485 the Application to Register Permanent Residence or Adjust Status.