Matthew 5:17-20
17 Think not that I am come to destroy the law, or the prophets: I am not come to destroy, but to fulfil.
18 For verily I say unto you, till heaven and earth pass, one jot or one tittle shall in no wise pass from the law, till all be fulfilled.
19 Whosoever therefore shall break one of these least commandments, and shall teach men so, he shall be called the least in the kingdom of heaven: but whosoever shall do and teach them, the same shall be called great in the kingdom of heaven.
20 For I say unto you, that except your righteousness shall exceed the righteousness of the scribes and pharisees, ye shall in no case enter into the kingdom of heaven.
In this passage, Jesus is specifically speaking of the law and keeping God's commandments. Jesus said He did not come to destroy the law but to fulfill it. To destroy something means putting an end to something through irreparable damage. To reiterate, Jesus is saying He did not come to put an end to the law, and then He said there will be consequences for breaking God's law, even for breaking the least of the commandments, and teaching others to do so will result in being the least in the kingdom of God. He then goes on to describe more consequences for disobeying God's law.
Jesus says that if a person's righteousness, that is, the righteousness in keeping God's commandments, is not greater than the righteousness of the scribes and pharisees in obeying God's law, they will not enter the kingdom heaven, meaning that person will not be saved. Jesus did not say keeping the law would save us, but He did say, however, if we fall short of keeping the law better than the scribes and the pharisees, the preachers and teachers of His time, we would not be saved.
The reason keeping the law cannot save us is because no one has ever been able to keep God's law perfectly as Christ did, fulfilling all the requirements of the law. To be saved we must repent of our sins, which means turning away from breaking God's commands and putting our faith in Christ to cleanse us from all unrighteousness, to lead and empower us to overcome sin and keep God's commandments. By doing this, we have entered into a covenant with God through Christ, and that covenant is called the new covenant. If we return to a life of sin, we have broken our covenant with God. Because He is full of mercy, He will chastise us and lead us back to repentance to keep our covenant with Him. However, if we continue in sin without repentance, He will eventually cut us off. Scripture says that God's Spirit will not always strive with man.