The phrase has it's origin in a conversation between Jesus and Nicodemus which was recorded at the beginning of chapter three of the Gospel of John:

Now there was a man of the Pharisees named Nicodemus, a member of the Jewish ruling council. He came to Jesus at night and said, "Rabbi, we know you are a teacher who has come from God. For no one could perform the miraculous signs you are doing if God were not with him."In reply Jesus declared, "I tell you the truth, no one can see the kingdom of God unless he is born again." How can a man be born when he is old?" Nicodemus asked. "Surely he cannot enter a second time into his mother's womb to be born!" Jesus answered, "I tell you the truth, no one can enter the kingdom of God unless he is born of water and the Spirit. Flesh gives birth to flesh, but the Spirit gives birth to spirit. You should not be surprised at my saying, 'You must be born again.' The wind blows wherever it pleases. You hear its sound, but you cannot tell where it comes from or where it is going. So it is with everyone born of the Spirit. (John 3:1-8)

 Being born again occurs at the moment when we recognize that we are a sinner and cannot save ourselves and so we ask Jesus to become our savior. At that moment, we are "born again" of the Spirit.  We are forgiven for all the sins we have ever committed or will ever commit because of Jesus' sacrifice on the cross. Peter also uses this phrase: 

Now that you have purified yourselves by obeying the truth so that you have sincere love for your brothers, love one another deeply, from the heart. For you have been born again, not of perishable seed, but of imperishable, through the living and enduring word of God. (1 Peter 1:22-23)

God is not content for us remain as we were. At the moment we accept Jesus, the Holy Spirit begins to dwell within us to make us a new person. Notice that before he speaks of begin born again, Peter talks about obeying the truth and loving deeply from the heart. The Holy Spirit dwelling in us begins to help us make those changes in our lives. With His help, we are able to love others more deeply and obey God's commandments more consistently. 

For some, this is an immediate change and their lives are completely transformed. I have seen people who were alcoholics or addicted to drugs immediately overcome those addictions through the power of the Spirit. For others, the process is slow and gradual. Eventually, we give up more and more of our old, sinful ways as the Spirit molds us into a new creation in the image of Christ.

Saying that someone is "born again" is simply another way of saying that they have accepted Jesus as their savior and as a result, they have been born again of the Spirit. Sometimes people refer to this as being "saved", or simply, "accepting Christ."

If you have not accepted Christ and been born again, we strongly encourage you to learn more about Jesus Christ and what he has done for you. Below this article, you will find some lines other articles which can help get you started. Please take a moment and discover these important truths about Jesus - it will be the most important thing you can ever do.