Here’s an expression that has the Pauline scholarly world in a stir: “works of the Law.” Let me break the options into two groups, with all sorts of variations overlooked, and suggest that how you understand this expression shapes how you understand nearly everything Paul says. Yep, it is that significant. What is your view?
First, there is what I will call the anthropological view: some think “works of the Law” describes the all-too-common human attempt to live in such a way that they are finally pleasing to God. Such persons do the Law to gain God’s favor, to maintain God’s favor, and to give themselves the sense that they are doing their best. The Reformation sabotaged this sort of yearning on the part of humans, and argued on the basis of Paul that there is nothing a human can do to be pleasing to God, that any such striving on the part of a human blows apart what God wants from humans (which is trusting God, not trying to do it on our own), and that this is the basal human yearning that stems from pride on the part of humans. Everyone, so they argued, is trapped into this yearning and that is only by a work of God’s grace that humans are relieved from this striving to find that they, like Luther, can find God’s grace and can trust God for everything, and that “works of the Law” are not something God’s people do. Instead, it is an expression that describes how non-trusting humans act before God to secure their own righteousness.

To one degree or another, the New Perspective on Paul (from Sanders to Dunn and Wright) contends that this is not at all what “works of the Law” is really about for Paul.
Second, for the community-marking view “works of the Law” refers to “human behaviors of obedience to the Law that separate Jews from Gentiles.” Works of the Law are boundary-markers (circumcision, food laws, purity laws, sabbath) between Jews and Gentiles (think Leviticus) and Paul sabotages any such boundary-marking by arguing (1) for faith and (2) a universal sense of sin and (3) a universal inclusion of all who have faith, Jew or Gentile, in the people of God.
Nothing is clearer from Romans than that this view has warrant in this text: Paul is out to show that the Jews cannot use the Law to prove their superiority to Gentiles. Why? Because the Torah shows that both Jews and Gentiles are sinners. This is not about trying to earn salvation but about identifying who is the true people of God and how that people is to live. (Now, I would not deny that Paul never trades in the anthropological argument, but I would say that “works of the Law” has this second sense.)
In other words, for this view (the New Perspective on Paul view) “works of the Law” is a Jewish problem, not a Gentile problem. Gentiles aren’t tempted to do “works of the Law” because they don’t have the Law. An example, in fact the only example, of “works of the Law” in Judaism is from the DSS at 4QMMT (4Q 399, C26-32):

Now, we have written to you 3 (C27) some of the works of the Law, those which we determined would be beneficial for you and your people, because we have seen that 4 (C28) you possess insight and knowledge of the Law. Understand all these things and beseech Him to set 5 (C29) your counsel straight and so keep you away from evil thoughts and the counsel of Belial. 6 (C30) Then you shall rejoice at the end time when you find the essence of our words to be true. 7 (C31) And it will be reckoned to you as righteousness, in that you have done what is right and good before Him, to your own benefit 8 (C32) and to that of Israel.

We can debate whether “works of the Law” refers to non-biblical rulings (this Qumran text) or to biblical rulings and laws, but the point remains the same: it refers to rules and actions that maintain strict separation between Jews and Gentiles and form the spiritual identity of the one who observes them. Further, these actions enable the one so practicing them to “other” those who don’t practice them. Now if you don’t see the potential value of this interpretation for modern church life, then let me make it plain as dirt: it would refer to any action that Christians do that intentionally separate them from others in order to foster their own identity and to deny that identity to others. I grew up with these “works of the Law”: movie going, drinking, smoking, and getting tattoos [thought I’d bring that one in today]. These acts enabled us to know we were in the right and all others were in the wrong. It is a safe world, after all, especially when you know you and your church mates are the only right ones in the whole dadgum world! The problem is that this is a form of self-righteousness (now I’m potentially slipping back into #1 but I’m keeping my head about me because it is not about anthropology but about social grouping).

I can’t tell you how much difference this makes to interpretation. For some it involves a near-crisis of theological understanding. Those who shift from #1 to #2 often have a very hard time then seeing that the instincts of #1 are hard to purge. And those in #2 go along their merry way and seem to be able to avoid nearly everything that the Reformers ended up arguing. If you think I’m exaggerating, listen in on the discussions at time. Right now some in the Reformed wing require a denial of #2 in theological exams for ordination.
Major warning: perhaps the most disappointing element of this discussion is that some in #1 have been super critical of #2. What’s more, at times they have (on their own) created the theology that those in #2 must believe. That is, they start accusing #2 folks of believing certain things (denial of imputation, etc) when those in #1 may never have said one word about this. It is foolhardy and uncharitable for those in #1 to create the theology of #2 and then to denounce them for denials of theological beliefs that those in #2 have never denied. The only way to know what another person believes is either to ask them or to read what they have said to see if they are saying what they are being accused of.