From the second chapter of the epistle to the Romans beginning in verse twelve

Paul has just concluded that God will judge the Jew and the Greek the same according to his or her actions. God will judge the wicked and glorify the righteous. Paul goes into a bit more detail.

¹² For all who have sinned without the law will also perish without the law, and all who have sinned under the law will be judged by the law.

Paul makes it more clear here that not having the law is no excuse for not following it. Though some do not have the law, they will still be condemned without the law. Those who did have the law will be held accountable for what sins they committed under the law.

¹³ For it is not the hearers of the law who are righteous before God, but the doers of the law who will be justified.

If simply having the law available to you were to justify you as righteous, then all the Jews would automatic saved. Paul argues that this is not the case. Paul says that the Jews would have had to actually keep (do) the law in order to be righteous by the law.

¹⁴ For when Gentiles, who do not have the law, by nature do what the law requires, they are a law to themselves, even though they do not have the law. ¹⁵ They show that the work of the law is written on their hearts, while their conscience also bears witness, and their conflicting thoughts accuse or even excuse them ¹⁶ on that day when, according to my gospel, God judges the secrets of men by Christ Jesus.

Paul strengthens the argument that knowing the law is not enough by pointing out that sometimes the gentiles keep the law all by themselves. He says "they are a law to themselves" meaning that their internal sense of right and wrong can condemn them. Paul comments that the law is written on the hearts of the Gentile, which is odd in light of the fact that the prophet Jeremiah promised that as a benefit of the New Covenant (Jeremiah XXXI.xxxiii). Paul does not mean to say that the gentiles already have the new covenant, but just that they can already tell right from wrong. Because of this, they can be judged when God judges us all.  

¹⁷ But if you call yourself a Jew and rely on the law and boast in God ¹⁸ and know his will and approve what is excellent, because you are instructed from the law; ¹⁹ and if you are sure that you yourself are a guide to the blind, a light to those who are in darkness, ²⁰ an instructor of the foolish, a teacher of children, having in the law the embodiment of knowledge and truth— ²¹ you then who teach others, do you not teach yourself?

Paul here explains that there are those who elevate themselves on account of them being Jewish. They think that they have the law, which qualifies them to teach the gentiles how to keep the law. Paul then asks them that as they are trying to teach the law to the gentiles whether or not they bother to teach themselves the law.

While you preach against stealing, do you steal? ²² You who say that one must not commit adultery, do you commit adultery? You who abhor idols, do you rob temples?

Paul keeps accusing the Jews here. He tells them they teach this, then ask if they actually break that law. 

²³ You who boast in the law dishonour God by breaking the law.

It is clear that Paul thinks they do break the law. He argues that the Jews are holding themselves up and saying "look at us we have the law" and yet they break the law. Because of this, they bring shame to God. Their hypocrisy is making the law essentially useless. 

²⁴ For, as it is written, “The name of God is blasphemed among the Gentiles because of you.”

Here Paul quotes something that resembles Isaiah LII.v, but does not match exactly. The ESV of that passage reads "Now therefore what have I here,” declares the LORD, 'seeing that my people are taken away for nothing? Their rulers wail,' declares the LORD, 'and continually all the day my name is despised.'" However, we have to turn to the NIV to find the word "blasphemed":
"'And now what do I have here?' declares the LORD. 'For my people have been taken away for nothing, and those who rule them mock,' declares the LORD. 'And all day long my name is constantly blasphemed.'" The point being that the Jews have not made a good impression of God among the Gentiles because of their hypocrisy.

²⁵ For circumcision indeed is of value if you obey the law, but if you break the law, your circumcision becomes uncircumcision.

Paul here refers to the act of circumcision, the sign of the covenant with Abram when God promised him many descendent. Paul here essentially ties the covenant with Abraham in its usefulness to whether or not the law is followed. It seems that Paul is saying that the foreskin can somehow be reattached in violation of the covenant if the Law is not followed. This is not at all what Paul means. He seems to be using the "second" meaning of circumcision here when the Lord commanded that Israel's hearts be circumcised. If this meaning is taken, Paul argues that the physical circumcision is annulled in the Jewish heart when the law is broken.

²⁶ So, if a man who is uncircumcised keeps the precepts of the law, will not his uncircumcision be regarded as circumcision?

Paul then asks a rhetoric question that follows the logic of the previous verse in reverse. 

²⁷ Then he who is physically uncircumcised but keeps the law will condemn you who have the written code and circumcision but break the law.

And Paul here confirms that the logic works by stating that it is the case that one who keeps the law, despite being physically uncircumcised, will be better off than those who break the law, despite being physically circumcised. 

²⁸ For no one is a Jew who is merely one outwardly, nor is circumcision outward and physical.

Paul kind of explains himself a bit by saying that nobody can be a Jew based on action alone, implying that there is an internal aspect as well. In the same way, circumcision is not just something to be done, but a truth to be applied to the heart.

²⁹ But a Jew is one inwardly, and circumcision is a matter of the heart, by the Spirit, not by the letter. His praise is not from man but from God.

Here it is established that being a Jew (and by extension circumcised) is truly a matter of the heart. As a result, it is God who rewards the circumcision.
I can not help but make a connection to the calling of David as the King when God has to explain to the prophet Samuel that God has chosen David, not by Samuel's standards of outward strength, but by God's standards which are based on the heart (I Samuel XVI.i-xiii).
In this section, Paul establishes that everyone has broken the Law, and thus sinned. He points out to the Jews that they have shamed God by boasting about the Law and then breaking it. He further argues that the Law and being a Jew (keeping the law) must be a matter of the heart.