For Christ is the end of the law for righteousness to everyone who believes. (ESV)
When you first look at this verse it is a little confusing to understand what Paul is talking about. Is he referring to the fact that Christ fulfilled the Law? In Matthew 5:17 Jesus says, "Do not think that I have come to abolish the Law or the Prophets; I have not come to abolish them but to fulfill them." I do not believe that is what Paul is referring to, although I am happy that Jesus fulfills the Law.
Paul has been making a point throughout this letter to the church at Rome that the new way to be made right with God is through the grace of God. It is a gift purchased for everyone who believes. The shed blood of Christ is the payment for all of our sin. The Law served to show us our inadequacy. God's righteous standard was unattainable by human efforts.
Christ is the end of the law for righteousness. If you want to be made righteous in God's sight you no longer need to try in your own efforts. This sounds like great news to you and me. It is good news. It is the essence of the Gospel. Why would anyone have a struggle with this truth if they truly wanted to be right with God?
The Jews had a hard time with this good news. They were used to relying on their own efforts to declare themselves holy. The Law appealed to their pride. When we rely on our own works it leads to pride; boasting. Paul says in Ephesians 2:8-9: For by grace you have been saved through faith. And this is not your own doing; it is the gift of God,  not a result of works, so that no one may boast.
If you are relying on your own efforts to be made right with God it can only lead to pride, but not to salvation. If you have not received the free gift of God I encourage you to do so right now!