This chapter, like all biblical chapters, has to be seen in the context of the whole Bible. The Bible teaches that we are to respect those who govern over us (regardless of how we feel about their policies or person). If, as Romans 13:1 says "there is no authority except from God" then those who are in authority are "God ordained" and deserving of our respect. However, this must be balanced with the other biblical passages that remind us that we must obey God rather than humans. In multiple places in the Bible, people are put in positions where they must disobey the governing authorities (see Acts 5:29) because to obey them would be unfaithful to God.
In the balance between these biblical positions are things like "civil disobedience" -- an act of disobedience made being willing to pay the consequences of that disobedience to highlight injustice or other structures of oppression. Note that civil disobedience (at least as taught and led by Dr. Martin Luther King and Gandhi) assumes the the disobedient one is willing to publicly pay the penalty for being disobedient. Even in these situations, however, we are not encouraged to "disrespect" to demean or to demonize those who are in authority over us. We are always called to pray for those in authority and to be good citizens. In the United States we are afforded another great responsibility in that we believe that government functions at the will and mandate of the people. As Christians we are called to pray and to discern and to cast our votes for the best people we can. As good citizens we pay our taxes and we obey the civil laws and authorities (except as it may be outlined above).
The chapter concludes with Paul's urgent appeal and reminder that time is fleeting and Jesus is returning and we should live honorably like the people of God we are and not be caught up in inappropriate or sinful behavior.