The greek word for "regeneration" in the New Testament is παλιγγενεσία (palingenesis). The word παλιγγενεσία had a primitive use by the stoics, most notably, Philo of Alexandria used to to refer to the rebirth of the world after it was destroyed by the flood.
Philo, The Life of Moses Vita Mosis 2.XII #65, (Greek text: Περι βιου Μωσεως) On which account everything was now made obedient to Noah; and even beasts, which up to that time had been savage, became gentle, and being tamed, followed him as their shepherd and superintendent; (62) and after they had all entered into the ark, if any one had beheld the entire collection, he would not have been wrong if he had said that it was a representation of the whole earth, containing, as it did, every kind of animal, of which the whole earth had previously produced innumerable species, and will hereafter produce such again. (63) And what was expected happened at no long period after; for the evil abated, and the destruction caused by the deluge was diminished every day, the rain being checked, and the water which had been spread over the whole earth, being partly dried up by the flame of the sun, and partly returning into the chasms and rivers, and other channels and receptacles in the earth; for, as if God had issued a command to that effect, every nature received back, as a necessary repayment of a loan, what it had lent, that is, every sea, and fountain, and river, received back their waters; and every stream returned into its appropriate channel. (64) But after the purification, in this way, of all the things beneath the moon, the earth being thus washed and appearing new again, and such as it appeared to be when it was at first created, along with the entire universe, Noah came forth out of his wooden edifice, himself and his wife, and his sons and their wives, and with his family there came forth likewise, in one company, all the races of animals which had gone in with them, in order to the generation and propagation of similar creatures in future. (65) These are the rewards and honours for pre-eminent excellence given to good men, by means of which, not only did they themselves and their families obtain safety, having escaped from the greatest dangers which were thus aimed against all men all over the earth, by the change in the character of the elements; but they became also the founders of a new generation (παλιγγενεσία), and the chiefs of a second period of the world, being left behind as sparks of the most excellent kind of creatures, namely, of men, man having received the supremacy over all earthly creatures whatsoever, being a kind of copy of the powers of God, a visible image of his invisible nature, a created image of an uncreated and immortal Original.
Clement of Alexandria also uses the rebirth of the world after the flood as the background for παλιγγενεσία (regeneration).
1 Clement 9: "(1) Wherefore, let us submit ourselves to his excellent and glorious will, and, becoming suppliants of his mercy and goodness, let us fall before him and betake ourselves to his mercies, having laid aside the vain toil and the strife and the jealousy that leadeth unto death. (2) Let us look steadfastly at those that have ministered with perfectness to his excellent glory. (3) Let us take as example Enoch, who, having been found just by reason of obedience, was translated, and his death was not found. (4) Noah, having been found faithful, preached, by his ministry, regeneration (παλιγγενεσία) unto the world, and by him God preserved the animals that entered with one consent into the ark.
Philo's usage was before the New Testament was written and is steeped in Stoic philosophy, and Clement is an apostolic church father who is writing as the New Testament is completing. The imagery is the same though: Noah, a righteous preacher is the new federal head of humanity, and the animal kingdom, and the metaphorically, the whole world. The flood has destroyed the earth, but Noah is the new head of humanity and reestablishes the earth. The earth has effectively been recreated through the destruction of the deluge (ie global flood). Philo has other uses of the word, but the Stoic idea (according to thayers) is the recreation through the earth by a fiery destruction. Clement turns to resurrection with the example of Enoch escaping death.
The baptism imagery is strong, and was the basis Augustine correlating regeneration (regeneratio) to baptism. This is also a theme in scripture as well that has caused the church to identify regeneration with baptism. For example, consider 1 peter 3:18ff and compare it to the idea of regeneration of the world in the flood, and noah's preaching to those who were destroyed by the waters, and salvation through noah's ark in the new creation.
1 Peter 3:18-22 ESV "18 For Christ also suffered once for sins, the righteous for the unrighteous, that he might bring us to God, being put to death in the flesh but made alive in the spirit, 19 in which he went and proclaimed to the spirits in prison, 20 because they formerly did not obey, when God's patience waited in the days of Noah, while the ark was being prepared, in which a few, that is, eight persons, were brought safely through water. 21 Baptism, which corresponds to this, now saves you, not as a removal of dirt from the body but as an appeal to God for a good conscience, through the resurrection of Jesus Christ, 22 who has gone into heaven and is at the right hand of God, with angels, authorities, and powers having been subjected to him."
Again, Jesus uses this same them of regeneration, by comparing the end to Noah's flood, where the unrighteous are swept away, yeilding to the new creation of the world:
Matthew 24:36-44 ESV 36 "But concerning that day and hour no one knows, not even the angels of heaven, nor the Son,  but the Father only. 37 For as were the days of Noah, so will be the coming of the Son of Man. 38 For as in those days before the flood they were eating and drinking, marrying and giving in marriage, until the day when Noah entered the ark, 39 and they were unaware until the flood came and swept them all away, so will be the coming of the Son of Man. 40 Then two men will be in the field; one will be taken and one left. 41 Two women will be grinding at the mill; one will be taken and one left. 42 Therefore, stay awake, for you do not know on what day your Lord is coming. 43 But know this, that if the master of the house had known in what part of the night the thief was coming, he would have stayed awake and would not have let his house be broken into. 44 Therefore you also must be ready, for the Son of Man is coming at an hour you do not expect.
(I will add, this is why suggestion Christians will be swept away in secret at the second coming (parousia), because the sweeping away was done by the flood in the washing of the world during Noah's day.)
Now, let's consider the Bible's use of "regeneration" (παλιγγενεσία). The word "regeneration" only appears twice in the bible:
Matthew 19:28 ESV Jesus said to them, "Truly, I say to you, in the new world (παλιγγενεσία), when the Son of Man will sit on his glorious throne, you who have followed me will also sit on twelve thrones, judging the twelve tribes of Israel.
Titus 3:5 ESV he saved us, not because of works done by us in righteousness, but according to his own mercy, by the washing of regeneration (παλιγγενεσία) and renewal of the Holy Spirit,
In Matthew 19:28, Jesus uses παλιγγενεσία in an eschatological context, refering to a future time when the twelve disciples will inherit twelve thrones and reign the twelve tribes of Israel respectively. The ESV translates παλιγγενεσία as "new world", and uses the word to refer to a future time when the world is recreated. Based on previous use, this implies a change of people, animals and the earth in the future.
In Titus 3:5, παλιγγενεσία is used in a past tense, referring to a recreation that has already happened to us. The reformed ordo salutis explained this the best by understanding anthropology as dichotomy: man is made up of two parts: the body and the spirit. The body will be regenerated at the second coming, but the past regeneration is of the spirit and corresponds to salvation. God elects men to be saved, not by any action of the individual, but a free choice of God and a gift of his mercy to individuals. A person is spiritually regenerated, due to god's choosing, and the spiritual renewal is accomplished by the Holy Spirit for those of God's choosing. At the coming of Jesus Christ, the receiving of his presence parousia, the body is then regenerated, as well as all the world.