I am a U.S. citizen, and my mother came to visit me in the U.S. on a visitor’s visa. Does she have to go back to her country in order to live with me here in the U.S. permanently?

No, this is not necessary. Your mother can adjust her status to permanent resident without leaving the country. You need to file a number of forms on her behalf, including an I-130, Petition for Alien Relative; an I-485, Application to Register Permanent Residence or Adjust Status; a G-325A, Biographical Information; an I-864, Affidavit of Support; as well forms I-765, Application for Employment Authorization; and an I-131, in order to obtain advance parole. This process may be quite complicated, so we recommend that you obtain qualified assistance from an immigration attorney or non-profit agency recognized by the Board of Immigration Appeals (BIA).