Romans 13:1-10 (KJV):

1 Let every soul be subject unto the higher powers. For there is no power but of God: the powers that be are ordained of God.
2 Whosoever therefore resisteth the power, resisteth the ordinance of God: and they that resist shall receive to themselves damnation.
3 For rulers are not a terror to good works, but to the evil. Wilt thou then not be afraid of the power? do that which is good, and thou shalt have praise of the same:
4 For he is the minister of God to thee for good. But if thou do that which is evil, be afraid; for he beareth not the sword in vain: for he is the minister of God, a revenger to execute wrath upon him that doeth evil.
5 Wherefore ye must needs be subject, not only for wrath, but also for conscience sake.
6 For for this cause pay ye tribute also: for they are God's ministers, attending continually upon this very thing.
7 Render therefore to all their dues: tribute to whom tribute is due; custom to whom custom; fear to whom fear; honour to whom honour.
8 Owe no man any thing, but to love one another: for he that loveth another hath fulfilled the law.
9 For this, Thou shalt not commit adultery, Thou shalt not kill, Thou shalt not steal, Thou shalt not bear false witness, Thou shalt not covet; and if there be any other commandment, it is briefly comprehended in this saying, namely, Thou shalt love thy neighbour as thyself.
10 Love worketh no ill to his neighbour: therefore love is the fulfilling of the law.

Power and Authority:

The first question which comes to mind when reading this passage is what exactly is a power? Many people make the mistake of assuming that anyone who claims to be a power is one, but the passage itself says otherwise. In verse 3, we are given the definition of a "ruler" or "power" as
"For rulers are not a terror to good works, but to the evil". In essence, anyone who is a terror to good works is not a God ordained power or ruler. This clarifies verse 1, which we can understand with the context of "the only source of true power (or authority - as it says in Young's Literal Translation) is God, and you have to meet his requirement in order to receive it". To ensure that our understanding of the verse is correct, we will compare it to the rest of scripture.

Biblical Obedience:

Acts 5:29 is the classic definition of Biblical submission and obedience: as Christians, our allegiance is first to God and his laws.

Acts 5:24-30:
24 Now when the high priest and the captain of the temple and the chief priests heard these things, they doubted of them whereunto this would grow.
25 Then came one and told them, saying, Behold, the men whom ye put in prison are standing in the temple, and teaching the people.
26 Then went the captain with the officers, and brought them without violence: for they feared the people, lest they should have been stoned.
27 And when they had brought them, they set them before the council: and the high priest asked them,
28 Saying, Did not we straitly command you that ye should not teach in this name? and, behold, ye have filled Jerusalem with your doctrine, and intend to bring this man's blood upon us.
29 Then Peter and the other apostles answered and said, We ought to obey God rather than men.
30 The God of our fathers raised up Jesus, whom ye slew and hanged on a tree...

As we can easily see, when anyone who claims to be a power or authority conflicts with the laws of God, they are not to be obeyed. If Romans 13 is understood to mean that authorities are to be blindly obeyed it would conflict with this scripture - which proves that we are on the right track to understanding Romans 13.

Romans 6:15-18:
15 What then? shall we sin, because we are not under the law, but under grace? God forbid.
16 Know ye not, that to whom ye yield yourselves servants to obey, his servants ye are to whom ye obey; whether of sin unto death, or of obedience unto righteousness?
17 But God be thanked, that ye were the servants of sin, but ye have obeyed from the heart that form of doctrine which was delivered you.
18 Being then made free from sin, ye became the servants of righteousness.

This passage, although directly discussing the Grace we have in Christ, also provides strong implications for those who "just follow orders" from a tyrant declaring himself to be an authority - but with no just authority from God.

Hosea 8:2-4:
2 Israel shall cry unto me, My God, we know thee.
3 Israel hath cast off the thing that is good: the enemy shall pursue him.
4 They have set up kings, but not by me: they have made princes, and I knew it not: of their silver and their gold have they made them idols, that they may be cut off.

If every ruler is automatically ordained by God, how could God "not have known" the princes that Israel set up? If everyone who claims to have power and authority really is a "minister of God" who is always executing His will, why do nations war with each other so often?

In closing, from Shadrach, Meshach, and Abednego, to Daniel, to Elijah, to John the Baptist, to Paul, and to the millions of other Christian heroes and martyrs, we are provided with many examples of disobedience to false authorities being obedience to God.

Cultural Implications:

In America, the supreme civil authority - the law of the land - is not a person. It is our Constitution. And the Constitution of the United States of America is in perfect harmony with the laws of God - even if certain elected officials aren't.

Thus, as Americans, we (and our elected officials) can read Romans 13:1-5 as:

1 Let every soul be subject unto the Constitution. For there is no Constitution but of God: the Constitution (and respecting officials) that be are ordained of God.
2 Whosoever therefore resisteth the Constitution, resisteth the ordinance of God: and they that resist shall receive to themselves damnation.
3 For the Constitution is not a terror to good works, but to the evil. Wilt thou then not be afraid of the Constitution? do that which is good, and thou shalt have praise of the same:
4 For the Constitution is the minister of God to thee for good. But if thou do that which is evil, be afraid; for the Constitution beareth not the sword in vain: for the Constitution is the minister of God, a revenger to execute wrath upon him that doeth evil.
5 Wherefore ye must needs be subject, not only for wrath, but also for conscience sake.