What exactly does it mean for one to be "born 'again"? The word "again" has a specific meaning.

Since words mean something, it is obvious that only one who has been born of God's doing a 'first time' can be 'born again.'

You will recall that Abraham and his wife Sarah, who was "dead in here womb" miraculously had a son who was promised by the Lord and through the Lord's intervention. She conceived and bore a "son of promise," Isaac.

Romans 9:7-9 (KJV)
7 ... the seed of Abraham, ... but, In Isaac shall thy seed be called. 8 That is, They which are the children of the flesh, these are not the children of God: but the children of the promise are counted for the seed. 9 For this is the word of promise, At this time will I come, and Sara shall have a son.

By this birth of Isaac, all "the children of Israel," as the lineal sons of Abraham, and his miracle son, Isaac, have been 'born of God's doing' a 'first time.' Lineally, this first birth of the children of Israel was through the God-given miracle birth of Isaac. So we can say "the children of Israel"  have been 'born of God's doing' a first time by this intervention of God to birth a people for His namesake.

We have record in the Gospels that "the children of Israel," Jesus concurring, made claim to be the sons of Abraham as his "seed" (Luke1:55, 73; John 8:56). This then explains why it is that Jesus declared to Nicodemus that he (Nicodemus) and also all the children of Israel needed to be 'born again'... this time spiritually.

The proper grammar of the KJV helps us to see the importance of Jesus' words.

7 Marvel not that I said unto 'thee' (Nicodemus, 1st person singular), 'Ye' (plural, "you all as children of Abraham and Isaac") must be born again. (John 3:7)
6 That which is born of the flesh is flesh; and that which is born of the (Christ) Spirit is (your) spirit. John 3:6 (KJV)

Thus then, the Gentiles who were not born of God a first time via Abraham cannot be born again. Gentiles must be spiritually born of God a first time. This explains why Paul, as "THE Apostle to the Gentiles," never uses the term "born again," choosing rather to be more accurate by using the word "regeneration" in (Titus 3:5).

5 Not by works of righteousness which we have done, but according to his mercy he saved us, by the washing of 'regeneration,' and renewing of the Holy Ghost; Titus 3:5 (KJV)