I was recently asked by someone on Facebook for a brief summary of what it means to be "born of the Spirit." Here is my response....
...the phrase "born of the Spirit" comes from John 3 when Nicodemus visits Jesus Christ and Jesus tells him that no one can see the kingdom unless he is "born again".
3:1 Now there was a man of the Pharisees named Nicodemus, a ruler of the Jews. 2 This man came to Jesus  by night and said to him, “Rabbi, we know that you are a teacher come from God, for no one can do these signs that you do unless God is with him.” 3 Jesus answered him, “Truly, truly, I say to you, unless one is born again  he cannot see the kingdom of God.” 4 Nicodemus said to him, “How can a man be born when he is old? Can he enter a second time into his mother's womb and be born?” 5 Jesus answered, “Truly, truly, I say to you, unless one is born of water and the Spirit, he cannot enter the kingdom of God. 6 That which is born of the flesh is flesh, and that which is born of the Spirit is spirit.  7 Do not marvel that I said to you, ‘You  must be born again.’ 8 The wind  blows where it wishes, and you hear its sound, but you do not know where it comes from or where it goes. So it is with everyone who is born of the Spirit.”
Jesus is basically saying that Nicodemus should know that the only way anyone is made "right" and "acceptable" to God is not by morality and by keeping the 10 commandments (since we've all broken them) but by receiving new life from Jesus himself.
1 Peter 1:3 says, "According to his great mercy, he has caused us to be born again to a living hope through the resurrection of Jesus Christ from the dead,"
So, basically to be born again (or "born of the Spirit") means to be given new life from Jesus--a "living hope" in Jesus Christ through the Holy Spirit who comes to live in us. It means that you have a living and loving relationship with Jesus by the Holy Spirit.