And you, being dead through your trespasses and the uncircumcision of your flesh, you, I say, did he make alive together with him, having forgiven us all our trespasses; having blotted out the bond written in ordinances that was against us, which was contrary to us: and he hath taken it out that way, nailing it to the cross; having despoiled the principalities and the powers, he made a show of them openly, triumphing over them in it (Colossians 2:13-15 ASV).

I am quoting from the American Standard Version (1901) because it translates the verses more precisely without too much interpretive glossing. 

Some have argued that what was cancelled, taken away, and nailed to the cross was simply the curse of the Mosaic Law. Others have argued that it was just the ceremonial aspects of the Mosaic Law. I don't think either of these are correct.

On the first view Paul would not have taken the trouble to delineate the Law as "written in ordinances" if he merely wanted to address the curse of the Law. He does this again in Ephesians 2:15,

Having abolished in the flesh the enmity, even the law of commandments contained in ordinances; that he might create in himself of the two one new man, so making peace;

Clearly Paul wanted to specify that what was cancelled was not simply the curse of the Law, but every provision written in the Law. This would include the decalogue (i.e 10 Commandments) and the 605 other commandments that applied these ten to various situations. In Ephesians, it was these very provisions in the Law that separated Jews from Gentiles, not the curse. As Paul would write, the uniqueness of the Jews was that they were entrusted with the Oracles of God (Romans 3:3). But Jesus took these away, making the two groups one in Him. 

What is more, the context of Colossians suggest that more than the curse is in view. In the very next verses Paul writes,

Therefore do not let anyone judge you by what you eat or drink, or with regard to a religious festival, a New Moon celebration or a Sabbath day. These are a shadow of the things that were to come; the reality, however, is found in Christ (Colossians 2:16-17).

These are all provisions in the Law of Moses. Paul tells the Colossian believers not to let anyone judge them on the basis of these specific provisions of the Law. In what sense would they be judged? They would be judged either for neglecting to follow these provisions or else for not following them as specifically proscribed in the Law. Thus, these very provisions are what was cancelled, taken away, and nailed to the cross. On this basis Paul can tell the Colossian not to let anyone judge them by the Law.

Now, some have suggested that Paul only had the ceremonial ordinances in view, particularly since these are the only ones he mentions in the following verses. However, the problem with this, is that the Bible never separates the Law in this way, namely, as moral laws, ceremonial laws and civil laws, though academics do this. The Bible always treats the Law as a whole. In fact, James would argue that breaking one part of the Law is like breaking all of it:

For whoever keeps the whole law and yet stumbles at just one point is guilty of breaking all of it (James 2:10).

What is more, the Jewish distinctive was not simply in the ceremonial laws but the whole body of laws given to Moses, especially the decalogue, which was written by the "finger of God":

And what other nation is so great as to have such righteous decrees and laws as this body of laws I am setting before you today? (Deuteronomy 4:8).

Thus, in the parallel text of Ephesians 2:15, the whole law is in view, of which Paul wrote, "having abolished in the flesh the enmity, even the law of commandments contained in ordinances." 

As I have argued elsewhere (e.g. in Galatians regarding circumcision), Paul probably highlighted certain "non-moral" laws in his letters in order to avoid confusion with the Law of Christ, which shares most of the same moral prohibitions as the Mosaic Law. 

Now, there are other verses that make clear that the whole Mosaic Law, even the entire Old Covenant, has been made to no effect and replaced by the New Covenant in Christ:

For if there had been nothing wrong with that first covenant, no place would have been sought for another... By calling this covenant “new,” he has made the first one obsolete; and what is obsolete and outdated will soon disappear (Hebrews 8:7-13).

But now, by dying to what once bound us, we have been released from the law so that we serve in the new way of the Spirit, and not in the old way of the written code (Romans 7:6).

He has made us competent as ministers of a new covenant—not of the letter but of the Spirit; for the letter kills, but the Spirit gives life. Now if the ministry that brought death, which was engraved in letters on stone, came with glory, so that the Israelites could not look steadily at the face of Moses because of its glory, transitory though it was (2 Corinthians 3:6-7).

Therefore, there is now no condemnation for those who are in Christ Jesus, because through Christ Jesus the law of the Spirit who gives life has set you free from the law of sin and death. For what the law was powerless to do because it was weakened by the flesh, God did by sending his own Son in the likeness of sinful flesh to be a sin offering. And so he condemned sin in the flesh, in order that the righteous requirement of the law might be fully met in us, who do not live according to the flesh but according to the Spirit (Romans 8:1-4).

In Acts 15 at the Jerusalem Council, the disciples settled this entire matter, concluding that the Jews had no business insisting on the Gentiles keeping the Law:

Then some of the believers who belonged to the party of the Pharisees stood up and said, “The Gentiles must be circumcised and required to keep the law of Moses. The apostles and elders met to consider this question. After much discussion, Peter got up and addressed them: “Brothers, you know that some time ago God made a choice among you that the Gentiles might hear from my lips the message of the gospel and believe. God, who knows the heart, showed that he accepted them by giving the Holy Spirit to them, just as he did to us. He did not discriminate between us and them, for he purified their hearts by faith. Now then, why do you try to test God by putting on the necks of Gentiles a yoke that neither we nor our ancestors have been able to bear? No! We believe it is through the grace of our Lord Jesus that we are saved, just as they are.” (Acts 15:5-11)

Therefore, let us understand, what was cancelled, taken away, and nailed to the cross was the whole Old Covenant Law of Moses. Therefore, Christians are no longer bound by the provisions of the Mosaic Law. Instead, Christ, through His blood, ratified a New Covenant, which is variously called "the Law of Christ" (1 Corinthians 9:21), "the law that gives freedom" (James 1:25), and "the new way of the Spirit" (Romans 7:6). The provisions of the Law of Christ are what we find in the New Testament letters. One might even see this as the New Covenant equivalent to the Books of Moses (Exodus - Deuteronomy), which contained the provisions of the Old Covenant Mosaic Law. But the most signficant provision of the New Covenant is that this law is written on our hearts, so that even if there is an imperfect practical outworking of its provisions, we, nevertheless, get the blessing of perfect obedience to it (cf. Jeremiah 31:31-34; Hebrews 8:8-12; 2 Corinthians 3:4-18; 5:16-21).