There is a massive conflict between obedience to God and obedience to the rulers that God has placed over us. Obviously, as noted above in #3, this came to be a particular thorny issue at the time of Paul's writing of Romans, as authorities and rulers were persecuting Christians. Thus the issue of how to respond as Christians when your God-appointed rulers were demanding you to recant your faith became of particular importance. Romans 13 is key in presenting obedience as part of Christian duty, as long as you recognise the higher authority of God on your life.
Obeying laws is something God expects His people to do (Exodus 20:6, for example: "But if you love me and obey my laws, I will be kind to your families for thousands of generations."). There are lots of references to obeying God's laws. As long as the laws of man don't directly contradict those, we are to obey the laws of our land.
In Peter 2:13-16, people are to submit themselves to the ordinances of man, whether he is king or governor. According to this passage, it is the will of God for people to obey, for by doing so they may put to silence "the ignorance of foolish men"-----
Another common New Testament story is that of paying taxes. In Matthew 22 or Luke 20 you can find the story of the teachers of the law and chief priests trying to trap Jesus after he has spoken against them in a parable. Faking interest they are completely disingenuous when they ask him if it is right to pay taxes to Caesar. Jesus has them take out a coin and identify the face on it. They (of course) say that it is the face of Caesar. Jesus says (to paraphrase), then give to Caesar what is Caesar's and give to God what is God's.
These verses are commonly used in sermons about civic duty and stewardship. Christians believe it is part of God's law to obey the laws of the government.
If you read Romans, Chapter 13, you will find an entire chapter about obeying the laws of man. Essentially, this chapter in Romans explains that man is to obey those that are put in authority over him, because those authorities are established by God, who is in charge of all things anyway. It explains that men who are in power on earth only hold their power because God allows it, so we are to obey those that govern us because in doing so, we are obeying God.
That said, however -- we are NOT to obey man when his laws conflict with God's laws. So, this chapter in Romans is NOT a justification, for example, for obeying Hitler, because he was an authority figure. Hitler's laws contradicted God's laws, and when this happens, man must obey the higher authority. If we were commanded by our authorities to do something that conflicted with God's laws, then we cannot obey man's laws.
The problem with this is that people try to pervert it. For example, it is not OK to blow up abortion clinics or kill abortion doctors just because one is opposed to abortion and believes that abortion is murder and God forbids it. We are to leave vengeance to God ("vengeance is mine," says the Lord, "I will repay.") God gets to define what he means by "eye for an eye, tooth for a tooth," not us.