Paul wants to preach the gospel in Rome; his calling to do that makes him a debtor both to Jews and to Gentiles. But, he knows what is there awaiting him: he knows the might of Rome. Still, Paul says, “I am not ashamed of the gospel…”. He knows that the prison in Rome is half-way up the Capitoline Hill and its entrance overlooks the Roman Forum. Still, he’s not ashamed nor is he afraid.
Romans 1:16-17 can be understood as the theme of the entire letter, though I’m not one to get too happy about such categories. The verses are thematically central. Here they are:

Rom. 1:16 For I am not ashamed of the gospel; it is the power of God for salvation to everyone who has faith, to the Jew first and also to the Greek. 17 For in it the righteousness of God is revealed through faith for faith; as it is written, “The one who is righteous will live by faith.”

I make three observations:
First, for Paul the gospel itself is the power of God, that power moves in the direction of salvation, and that salvation is for anyone/everyone/ anyoneyoucanfind who trusts in Jesus Christ.
Second, that gospel brings about the righteousness of God. I don’t know how to read 1:17 as anything other than some kind of explanation of 1:16. Paul says the gospel is the power of God to save anyone who believes, and now he says it is the righteousness of God that is being revealed from faith to faith. Gospel for salvation and righteousness of God are near synonyms at the least.
Third, let me do my best to avoid a dissertation and simply explain what “righteousness of God” means. If you’ve heard or not heard, this expression is the center of a huge debate and is smack-dab in the middle of those debates between classical Reformed thinking and the New Perspective on Paul. Some in the former group have not taken the time to read the appropriate studies or combed through the Jewish evidence, and some in the latter seem to have no regard for what Luther and Calvin fought for. They’d all do better to do a little listening.
In general, we can say God’s righteousness can refer to an attribute of God (God is righteous), a status given to humans by God’s grace (we are made right with God), or an activity of God (God’s saving justice at work to make things right).
Now, here’s what is going on as I see it: the Reformed thinkers contend that the righteousness of God is God’s holiness and justice being honored and preserved as God brings the world’s moral scale back to even through the death of Jesus Christ — a death that is at once a propitiation of God’s wrath and an expiation of sin. Hence God’s holiness works it out that humans can be declared “right with God” (the righteousness that comes from God) by God’s grace. Hence, righteousness here is a gift and a standing. And this focus is singularly individualistic: the focus is on how individual believers might be made right with God.
The New Perspective folks don’t necessarily deny any of this, but they see the balance of interest on Paul’s part to be elsewhere. The righteousness of God is God’s action to make things right in this world through Jesus Christ (life, death, resurrection) and the saving benefits that are granted to those who are in Christ Jesus. They see a corporate dimension, an individual dimension, and both are taken up into God’s transforming and world-changing power at work here and now.
Many today — the wise we might say — think there is no reason to choose between three obvious dimensions of an expression. Hence, my former colleague (whose son’s basketball team in high school always seemed to beat my son’s, which means my quoting him here is a peace offering), Doug Moo, defines it like this: “the act by which God brings people into right relationship with himself” (Romans, 74; had we won I would have made that title a link to amazon.com!).
I would slightly modify Doug’s definition and make it a little less individualistic: the act of God, shaped as it is by God’s attribute of being righteous, whereby he makes things right in this world. Here is where the Reformed thinking and the New Perspective varies: for many in the former camp this “right-making” by God is nothing more than a “right-declaring” and not a morally transforming “right-making.” This, of course, brings in the age-old and dog-eared categories of “imputation” vs. “impartation,” and I’d really like to avoid using those terms too much.
But I would say this: the gospel is the power of God for salvation, and the righteousness of God is another way of describing that gospel, and I’d like to suggest that Paul could not have imagined a being made right with God that did not also mean a being made right in the sense of being flooded with God’s Spirit so that God’s will was made manifest. The Reformed might like to distinguish these two acts (justification, sanctification) in an ordo salutis, but I doubt very much that Paul made such distinctions. To be saved is to be saved — how’s that for a clever line?! In Paul, to be “saved” is to be given life and to be removed from death; to have life is to live in the newness of life.
Now one of the central features of God’s “right-making” is that God joins into one family both Jews and Gentiles, and I see this as very important to what Paul is saying in Romans. So, we might say that the righteousness of God is a big term for God’s making things right in and through Jesus Christ in the here and now as an anticipation of the there and then. When emerging folk speak of “walking in the way of Jesus” they are, if one has a mind to see it, talking about justification.