A cursory reading of these verses suggests that a prayer offered in faith for the healing of a sick person will result in God restoring him/her. Of course, there is a real difficulty reconciling what James says with the experiences we've all had in which a congregation has prayed fervently for someone's healing only for that person to die. Is James wrong? Are our prayers not offered in faith? Why does God choose to heal some and allow others to die?
I think these two verses must be read in the context of the verses around them and with a mind to Christ's life and teachings. In verses 10-11 James encourages his readers to persevere in suffering and trust that God will ultimately reward them, reminding them of the prophets and Job. I'd be hard-pressed to think of an Old Testament prophet who didn't reap persecution and hardship for doing God's work--I'm reminded of Hebrews 11:13. As for the story of Job, although God rewards Job in the end, He never offers Job an explanation for his suffering but reminds Job that He is sovereign and His ways are often inscrutable to man. Both examples reinforce our humility before God, and how faith requires us to submit to His will, not to conform Him to ours. Of course, the ultimate example of submitting to the Father's will is in Christ's prayer before His betrayal--"not my will, but Thine be done." In studying this verse I was reminded of the movie Shadowlands in which Anthony Hopkins as C.S. Lewis explains to a friend that he believes prayer has more to do with changing him than changing God's mind--I believe Lewis expands on that in A Grief Observed.
However, I think ultimately James is more concerned with spiritual healing than physical. In verse 16 the healing he speaks of seems spiritual in nature given the charge to Christians to confess their sins to each other and pray for each other.