Paul has used a question/answer style earlier in Romans to raise possible objections to his way of thinking and then to answer them.  He resumes this approach in Romans 6. Romans 6 and 7 deal with one of the main objections to his teaching. He mentioned it back in 3:8. People are parodying him as if he were teaching, "Let us do evil that good may result."
So Paul hits this false accusation head on. "Shall we go on sinning so that grace may increase?" (6:1). "Shall we sin because we are not under law but under grace?" (6:15). Paul's answer in both cases is an emphatic "no": "By no means!"
In truth, no mainstream interpreter of Paul has ever accepted the old parady of Luther, "Sin boldly that grace may come." Luther never made this statement, and you will not find support for such an approach to sin from any mainstream Lutheran, Reformed, Baptist, Methodist or any other major Reformation tradition. Christian groups may differ in their definitions of sin and they may differ in how much they think a Christian will normally sin, but no tradition worthy of the name Christian will teach that Christians will continue to sin no differently than before they received the Holy Spirit.
What is confusing to so many readers is the role that Romans 7:13-25 plays in Paul's argument. So many us identify with the words in 7:19: "what I do is not the good I want to do; no, the evil I do not want to do--this I keep on doing." Perhaps even the majority of Christian readers of Paul today latch on to this sentiment because it fits with their own personal experience and then they ignore completely the context in which Paul makes it. It is perhaps the single greatest misunderstanding of Paul's writings, as common as it is completely wrong.
In context, Paul's argument has pushed him to his single most vulnerable point. What was God doing with the Jewish Law all those years if in fact, as Paul is arguing, it is not an effective path to God? Paul was saying that Gentiles did not have to observe the parts of it that were Jew-specific, things like circumcision. And he was saying that Jews were not able to be right with God on the basis of how well they kept it. But he also says that getting right with God on the basis of his grace administered through Christ is no excuse to sin, to "do wrong," which of course was defined more than anywhere else in the Law.
This line of thinking is confusing to us and it may have also been not a little confusing to the Romans as well (cf. 2 Pet. 3:16). So what is the Law good for, Paul? You say we are not under it but you say we do not violate it. You say it was never effective but then why did God institute it in the first place? It is these sorts of questions that lead to Romans 7, where Paul explains the role of the Law in God's plan and how that applies to believers.
I am convinced that a great deal of our confusion comes from the fact that Paul glides seemlessly between a number of different meanings for the word "law" in these passages...
tomorrow: definition of sin, different uses of the law
 Usually called a "diatribe" style.