to rule, to be first in authority over others
In the form that means ruling, archo carries the ancient idea of holding first place and exercising governing authority. This use of the verb appears only twice in the New Testament, both times describing rulers who wield power over the nations.
In Mark 10:42 Jesus contrasts the way gentile rulers exercise this authority (lording it over people) with the servant leadership he calls his disciples to. In Romans 15:12 Paul cites Isaiah's prophecy of the root of Jesse rising to rule the nations. Both uses carry the political, royal sense of rule that this form of the verb held throughout classical Greek.