If an individual does not have a green card, and either overstayed a period of authorized admission or entered illegally and cannot adjust status in the US, there is still an opportunity through cancellation of removal to stay in the US and obtain the status of a lawful permanent resident and obtain a green card.
Cancellation of removal usually is a defensive move taken by someone about to be deported, occurring after a person has been put into removal proceedings. This implies a Notice to Appear has been issued demanding the appearance before an immigration Judge. Basically, if the foreign national, is a “respondent” in court, may ask for relief from the court in the form of a cancellation of removal application.