As to who exactly that means, it might mean the "Old Stock" Americans, who as I said of Neil Gorsuch's ancestors, were already present in the US, meeting the boats that came in to Ellis Island in the 1890s, but we usually use it to mean "the Historic American Nation as it had evolved up to 1960"—I. E. the "Old Stock" plus the Ellis Island contingent, who the Great Pause had encouraged to assimilate.

It would be easier to say who it doesn't mean—people like Filipino illegal immigrant Jose Antonio Vargas, the Arab taxi drivers who shut down JFK recently, people like that.