This passage has been one of the bullet points--on either side of the argument-- of those who debate the proper role of the Church in politics today. Taken in isolation, it could seem to provide ammunition for the side that argues against the Church being involved; after all, "those authorities that exist have been instituted by God." Also, taken by itself, it could be seen as an injunction to believers to yield to civil authority, almost without question of resistance.
Needless to say, there is plenty else in Romans to balance the perspective offered here, and to view it in proper context. For the main thrust of Paul is to argue that Jesus is Lord, and the Emperor Caesar is not Lord. Indeed, the cult of emperor worship throughout the first-Roman empire was a religion that tended to foster civil obedience and therefore a continuance of the order, justice and discipline that at first was imposed by the sword. Paul is saying that the emperor is not God, but is himself subject to the true God. The civil order is useful for temporal purposes, not for eternal.
At a time when Christanity contended with the empire, Paul was therefore saying: walk the middle road. Obey the just laws of the empire. But worship the God who alone can bring truth and ultimate justice.