For follow-up discussion and general commentary on the topic. Comments are sorted chronologically.
It means that Christians are not obligated to do the law as a means of trying to obtain right standing with God. Righteousness does not come by the law, but only through Jesus Christ our Lord. Keeping the law will never make one righteous, because no one is able to keep the whole law perfectly. If you miss it in one area, you are guilty of breaking the entire law. How frustrating is that!
I agree with Mr Houndaman. Yes Jesus fulfilled the law so that everyone who accepts Him as their savior will be with Him in heaven.
Jesus said that the greatest commandment is to Love God with all your heart and love your neighbor as yourself. It is only through the power of the Holy Spirit that we can do this.
As Paul states our sin nature wars with the Holy Spirit daily thus the need to pick up our cross daily. I interpret this to mean battle our nature (the cross we must bear) by leaning on the Holy Spirit who strengthens me.