In the Lutheran confession of Christianity, confirmation is closely connected with Baptism. It is a re-affirmation of God's promises made to an individual when he was baptized (i.e. "confirmed").
Confirmation has its roots in the early New Testament Church when most people who became Christians were adult converts. At the vigil of Easter, the convert (after going through months or years of catechesis, or instruction) would be baptized by the pastor. The baptismal ceremony involved an exorcism (i.e. a casting out of the devil), a renunciation of the "devil and his ways," and a confession of faith using the three articles of the Apostles' Creed. Following the convert's baptism, the bishop would "confirm" the baptism.
As the Church grew and more and more infants were baptized, the order of things began to be reversed. First, the child was baptized, then catechized (i.e. taught the faith), then confirmed. Yet, the rite of confirmation was still intimately connected with Baptism, although separated by time. Thus, confirmation is seen as an affirmation of God's promises made to the baptized.
Regarding Martin Luther, he spoke of being "born again" in the context of finally understanding the verse in Romans 1:16-17: "For I am not ashamed of the gospel, for it is the power of God for salvation to everyone who believes, to the Jew first and also to the Greek. For in it the righteousness of God is revealed from faith for faith, as it is written, “The righteous shall live by faith.”
In the preface to his collected works, Luther writes that he had struggled with the concept of God's righteousness and what this verse meant. He had taken the verse to mean God's righteous judgement against sinners, and this terrified him and actually caused him to hate God. While teaching a class on Romans, though, he finally understood that the righteousness of God is His righteousness by which He accounts us sinners as righteous due to His grace for the sake of Christ. Thus, the Gospel of Jesus Christ gives us Christ's righteousness as a gift; this is why it is "the power of God for salvation to everyone who believes."
This "discovery" of the Gospel caused Luther to rejoice and feel "born again." He rejoiced because he finally understood that God is giving us Christ's righteousness as a gift, apart from any works or merit of our own (see http://strangetriumph.wordpress.com/2011/06/24/martin-luther-on-the-righteousness-of-god/ for the whole of Luther's quote). Luther was not speaking in the modern evangelistic sense of being "born again." Rather, he meant that he finally understood God's grace which is given to us freely on account of Christ's death and resurrection.
Lutherans do not typically speak of being "born again." If pressed, though, we would typically say that we were born again in our Baptisms, because in our Baptisms God killed our old natures and raised us to new life in Christ (Romans 6:3-4).