ANSWER: Your question regarding the meaning of born again is a very important one. The scripture you are referring to is part of the conversation Jesus had with a Pharisee named Nicodemus. This Pharisee first approached Jesus at night (in order to conceal his meeting from the other religious leaders) to tell him they (the Pharisees) KNOW he came from God because no one else can do the same miracles he does (John 3:2)! Jesus interjects by stating what is required to enter the Kingdom of God.
" Verily, verily, I say unto thee, Except a man be born (Greek word gennao, Strong's Concordance #G1080) again . . . " (John 3:3, KJV)
Note that in John 3 there are three different types of births discussed. Nicodemus initially understood Jesus' statement to him (verse three above) in a literal sense. He thought Christ was referring to being born of a woman as all mankind has had to experience since Adam and Eve (verse 4). The Lord's response was that humans must come out of both water and spirit in order to enter into the Kingdom of God (verse 5).
In the world of Christianity the expression "born again" generally is used to refer to someone who has "accepted Jesus" and been baptized. Some say they believe in the doctrine of 'once saved, always saved' by this. It should be abundantly clear from the entire discussion between Nicodemus and Jesus in John 3 that there is a SECOND birth. Note Nicodemus asks if one can enter a SECOND time into his mother's womb. Jesus, however, is talking about a DIFFERENT womb.
The womb Jesus was referring to was the EARTH. He was discussing this womb when later, in John, he said the hour was coming when those who are dead and awaiting a resurrection will hear his voice and come back to life.
28. Do not wonder at this . . . all who are in the graves shall hear His voice 29. And shall come forth: those who have practiced good unto a resurrection of life . . . (John 5:28 - 29, HBFV).
One could say that a Christian who was symbolically buried in a watery grave at baptism (see Romans 6:3 - 4, Colossians 2:12 - 13, 3:9 - 10 and 2Corinthians 5:17) then resurrected from it has been reborn or BEGOTTEN (or "begun" as the Greek seems to indicate) of water, receiving the earnest of God's spirit for his mind, giving him the mind of Christ. However, we are most interested in the brand new start which will bring immortality to us. Jesus was born of the flesh as the seed of David. However, you will notice that he, as God's son, was born with POWER when he was resurrected from the dead.
3. Concerning His Son; Who came from the seed of David according to the flesh (he was born out of David's lineage), 4. Who was declared the Son of God with power, according to the spirit of holiness, by the resurrection from the dead - Jesus Christ our Lord (Romans 1:3 - 4, HBFV)
The three births referred to in John 3 are being born of the flesh (how we got into the world), of the water (symbolizing baptism and the receiving of God's spirit in order to live a life of overcoming) and of the spirit (a true Christian's resurrection from the dead).
We will not have been "saved" from death until we are born again with spirit bodies by a resurrection out of the womb of the earth, as John 3:6 states. Those composed of spirit can be invisible or visible (John 3:8). The power to make us spirit beings comes from above, from God. While there are many other scriptures showing the resurrection as the birth of a new body, I hope this brief answer aids your studies.