In Romans 10:4 the apostle Paul wrote that Christ is the end of the law. 

Rom 10:4  For Christ is the end of the law for righteousness to everyone who believes.

What does Paul mean by this? From such verses as Matthew 5:17-18, it should be clear that Jesus Christ did not destroy or abolish the law.

Mat 5:17  “Do not think that I came to destroy the Law or the Prophets. I did not come to destroy but to fulfill.
Mat 5:18  For assuredly, I say to you, till heaven and earth pass away, one jot or one tittle will by no means pass from the law till all is fulfilled.

So, what does Paul mean?  Let’s have a close look at this Romans 10:4 including the preceding verses for context.

Rom 10:1  Brethren, my heart’s desire and prayer to God for Israel is that they may be saved.

Note that Paul wants the nation of Israel to be saved.

Rom 10:2  For I bear them witness that they have a zeal for God, but not according to knowledge.

Israel has a zeal for God, but because they lack knowledge about God and what He wants, their zeal is not serving them well.

Rom 10:3  For they being ignorant of God’s righteousness [dikaiosune], and seeking to establish their own righteousness [dikaiosune], have not submitted to the righteousness [dikaiosune] of God.

The Greek word translated “righteousness” is dikaiosune.  It comes from the root word dikaios, meaning innocent, just, right.  Another meaning of dikaiosune is “justification”.  In fact, this verse makes much more sense if it is thought of in terms of justification rather than righteousness.  Justification is the process of being made right with God.  See “What is Justification?” for more details.  Out of ignorance, Israel was seeking to be justified their own way, without Jesus Christ.  Earlier in Romans, Paul makes it clear that we are justified by faith in Jesus Christ, and not by works of law.

Rom 3:28  Therefore we conclude that a man is justified by faith apart from [the] deeds of [the] law.

We now come to the verse in question.

Rom 10:4  For Christ is the end of the law for righteousness [dikaiosune] to everyone who believes.

Now that we know Paul is talking about justification, this verse makes more sense.  For those who are not ignorant and believe in Jesus Christ, Christ has put an end to the idea of using the law for justification.  More tersely, Christ is the end of using the law for justification, to everyone who believes.  Romans 10:4 can be translated more clearly as:

Rom 10:4  For Christ is the end of [using] the law for justification to everyone who believes.

Romans 10:2-4 can be roughly paraphrased as follows:
Previously, some in Israel had been very zealous for God, but ignorant.  They were ignorant of how God justifies us and sought to justify themselves (through the law without Jesus Christ).  For those who believe in Jesus Christ (and understand how they are justified), Christ has put an end to the idea of being justified through the law.

For related articles, see:
“What was the Galatian heresy?”
“What do “under law” and “under grace” mean?”