Regeneration is a popular topic among Calvinists. It is also an important topic, because it is generally agreed that one is not saved without regeneration, and it is crucial to the topic of grace. In fact, the topic can be pivotal to Calvinism and Arminianism.
R.C. Sproul, a Reformed Theologian, gives us a succinct definition of regeneration which I have no disagreement with: "Regeneration literally means "to regenerate again." It is the concept that rests upon Scripture's teaching concerning rebirth or being born anew."1 There is one passage relevant to this topic which uses the term "regeneration": He saved us, not on the basis of deeds which we have done in righteousness, but according to His mercy, by the washing of regeneration and renewing by the Holy Spirit. Titus 3:5. The force of this term carries with it the idea of a one time event where one is restored or reborn-- hence this is equivalent to the idea of being born again as we see it in John.
The critical debate of this concept, between Calvinists and Armenians, is when the event of regeneration occurs relative to the conversion from disbelief to saving faith. If regeneration is prior to faith, then regeneration is the driving force behind faith and thus faith isn't so much something you choose to do but what occurs to you (hence, this position leads inextricably to Calvinism's Irresistible Grace). On the other hand, if faith is prior to regeneration, Calvinists will have almost no grounds for Irresistible Grace. Unfortunately, most passages on the subject don't have any obvious chronological markers, so many theologians have tried to fill in the blanks.
Nevertheless, is there any Biblical indication which side is right? Indeed there is, though it'll take some digging in, but once again we are going back to a familiar passage. And this passage is John 3:14-15.
However a little context is needed. In the first section of John 3, Jesus tries to explain to a Pharisee named Nicodemus the concept of being born again (being born again, mind you, is equivalent to being regenerated; John even uses similar language as found in Titus 3:5).2 Nicodemus failed to understand (John 3:9-12), so Jesus went on to clarify in less elusive terms: "As Moses lifted up the serpent in the wilderness, even so must the Son of Man be lifted up; so that whoever believes will in Him have eternal life. Jesus continued explaining the importance of belief (or was it John? There are some interpreters who arena't sure where Jesus ends and John gives his own commentary in Viv. 16ff.),3 but for our purposes what is important is in this immediate nugget.
So, what's so significant about that passage? Consider that Nicodemus was a Pharisee, and Pharisees knew the whole Hebrew Bible (our Old Testament) like the back of their hands. When Jesus mentioned the event where Moses "lifted the serpent," Nicodemus would have recalled exactly what Jesus was talking about, including the whole background story behind it.
In Numbers 21, we read where Moses lifts up the serpent. Because of their contempt, God cursed the ancient Israelites by sending venomous snakes into their camp. Many people died because of them. The people repent to Moses, "We have sinned, because we have spoken against the LORD and you; intercede with the LORD, that He may remove the serpents from us." Numbers 21:7. However, God doesn't take the serpents away, but instead commands Moses, "Make a fiery serpent, and set it on a standard; and it shall come about, that everyone who is bitten, when he looks at it, he will live." Numbers 21:8. Moses made a bronze snake and put it on a pole, so anyone who was bitten could look up to it and be saved.
Jesus is drawing a comparison between the lifting of the bronze serpent and His own sacrifice (which is also on a pole of sorts; the "lifting up" language is more clear later on [John 12:32-33]). Just as those who looked up to the serpent for their salvation will live, so will those who look to the cross to be saved will be given eternal life.
So, what does this all have to do with Regeneration? It's in the parallelism. One must be born again to be saved. The looking to the serpent in Num. 7 resulted in a regeneration of sorts (that is, from being terminally poisoned to being healthy). Thus, if we look to the cross by faith, we will be regenerated, which will lead to salvation. In other words, one must come to faith in order to be born again, and those who are born again are saved.
Thus when we take into account the Hebrew background behind Jesus' explanation, it becomes clear how faith and regeneration relate to each other in salvation. The Israelites, in faith, looked to the serpent to be regenerated, and thus would live again. The believer, in faith, looks to Christ to be regenerated, and thus will live eternally. This is as contrary to the Calvinist doctrine on regeneration as it Scripture has been contrary to Calvinism in general.