Confess your faults one to another, and pray one for another, that ye may be healed. The effectual fervent prayer of a righteous man availeth much.
Another interpretation of this saying is, "The prayer of a righteous person has great power as it is working." In other words, prayer works. God hears all prayers, but the prayers of a righteous person who keeps the commandments and is in tune with the Holy Spirit get more attention because they are already in harmony with the Lord's will. They are easy for God to answer.
I do not claim to be a righteous person, or that my prayers are always fervent or effective, but I try to pray regularly for others, and I like to think that others are praying for me. I need the prayers of others to bolster my own pathetic prayers. Most importantly, I rely on the prayers of Jesus on my behalf:
"I pray for them: I pray not for the world, but for them which thou hast given me; for they are thine. And all mine are thine, and thine are mine; and I am glorified in them." (John 17:9-10)
"Listen to him who is the advocate with the Father, who is pleading your cause before him— Saying: Father, behold the sufferings and death of him who did no sin, in whom thou wast well pleased; behold the blood of thy Son which was shed, the blood of him whom thou gavest that thyself might be glorified; Wherefore, Father, spare these my brethren that believe on my name, that they may come unto me and have everlasting life." (D&C 45:3-5)
The only truly righteous person to have lived on the earth is Jesus, and surely His prayers are effectually fervent. Though I fail in many ways every day, I have hope because Jesus is praying for me.