The U.S. has a mutual agreement with some countries whereby citizens of both countries can enter the other country without a visa for short visits. For example U.K. citizens can come to the U.S. for up to 3 months without a visa. The visa waiver cannot be extended in the U.S. and cannot be converted to a green card even if an I-140 is approved. In such a case, the person would have to return to his or her home country and go through consular processing. For instance, if an applicant has a first preference green card application approved, his or her spouse and children will receive green cards at the same time as the applicant.
Citizenship is different than permanent residence.
Citizenship can be sought only after holding permanent residence for a specific amount of time. Citizenship confers all rights upon the successful applicant as are held by native born U.S. citizens.