This interesting question has always been a matter of debate, even if it's not as significant as many other controversies. What exactly happened to Jesus in between His death and resurrection? Some people say He went to Hell. Other say He went to Heaven. Still others say something else entirely. So, what's Biblical? There are a few verses to consider.
"For you will not abandon my soul to Sheol, or let your holy one see corruption." Psalm 16:10
"…he foresaw and spoke about the resurrection of the Christ, that he was not abandoned to Hades, nor did his flesh see corruption." Acts 2:31 (citing Ps. 16:10 about Jesus)
"And he said to him, 'Truly, I say to you, today you will be with me in Paradise.'" Luke 23:43
"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, in which he went and proclaimed to the spirits in prison…" 1 Peter 3:18-19
These verses give us three details about the events after Jesus' death. He was in Hades/Sheol, He was in "paradise," and He preached to "the spirits in prison." What does all this mean? Let's just go in order and examine each of these items.
First we see that Jesus was in Hades, or Sheol. What does that mean exactly? The answer is somewhat confusing. First, we must note that Sheol is not necessarily Hell. Strictly speaking, Sheol refers to the general abode of the dead, both righteous and wicked, and sometimes is used almost synonymously with "death" or "the grave." So Sheol technically includes Heaven and Hell, and that is apparent when righteous men are spoken of as going there (see Gen. 37:35, for example). However, Sheol was quite often used with the connotation of the destiny of the wicked, as is evident in verses such as the following:
"But if the Lord creates something new, and the ground opens its mouth and swallows them up with all that belongs to them, and they go down alive into Sheol, then you shall know that these men have despised the Lord." Numbers 16:30
"The wicked shall return to Sheol, all the nations that forget God." Psalm 9:17
"O Lord, let me not be put to shame, for I call upon you; let the wicked be put to shame; let them go silently to Sheol." Psalm 31:17
"Like sheep they are appointed for Sheol; death shall be their shepherd, and the upright shall rule over them in the morning. Their form shall be consumed in Sheol, with no place to dwell. " Psalm 49:14
"If you strike him with the rod, you will save his soul from Sheol." Psalm 23:14
Interestingly enough, Sheol was used with the same variety as "death" is in the New Testament, referring in almost equal proportion to physical death and spiritual death. So, based simply on Psalm 16:10, it is very difficult to ascertain if Sheol was being specific about where Jesus was, or was simply being used as a synonym for death. The latter seems likely, since Peter said the verse referred to the Resurrection, and so the focus would be Christ's physical death.
Acts 2:31 still makes for an interesting verse, though, because it comes directly after Peter quotes Psalm 16:10, except he uses "Hades" instead of "Sheol." What's the difference? Well, it actually varies. See, Hades was the Greek equivalent of Sheol. In fact, in the Septuagint (the ancient Greek translation of the Old Testament, also called the LXX), the word "Sheol" was consistently translated with "Hades." So at first glance Hades seems to have no impact on the debate. However, when one examines references to Hades in the Gospels (they are rare; Gehenna was more commonly used), they seem to focus almost exclusively on the destiny of unbelievers, Hell. For example:
"And you, Capernaum, will you be exalted to heaven? You will be brought down to Hades. For if the mighty works done in you had been done in Sodom, it would have remained until this day." Matthew 11:23
"The rich man also died and was buried, and in Hades, being in torment, he lifted up his eyes and saw Abraham far off and Lazarus at his side." Luke 16:22b-23
With such a small sample size, we cannot make any conclusions with certainty. Therefore, examining the references to Sheol and Hades in relation to Jesus' death doesn't tell us much more than that He was dead. It is possible Hades was meant to mean Hell in this context, but it cannot be proved.
The next thing to consider is that Jesus said the thief on the cross would be with Him that day in "paradise." Now, most people assume this means Heaven. Does it? There are only two other uses of the term "paradise" in the New Testament.
"And I know that this man was caught up into paradise—whether in the body or out of the body I do not know, God knows." 2 Corinthians 12:3
"He who has an ear, let him hear what the Spirit says to the churches. To the one who conquers I will grant to eat of the tree of life, which is in the paradise of God." Revelation 2:7
In 2 Corinthians 12:2, Paul says, "I know a man in Christ who fourteen years ago was caught up to the third heaven—whether in the body or out of the body I do not know, God knows." This indicates that paradise is the third heaven, which was the Jewish way of referring to Heaven where God lives (the first heaven was the sky, and the second heaven was the cosmos). Also note that the "tree of life," which is said to be in the "paradise of God." The tree of life is mentioned two other times in Revelation, and both refer to it in the New Jerusalem, final Heaven. It seems evident, then, that paradise is Heaven, and Jesus went to Heaven immediately after He died. This is also hinted by Jesus' statement in Luke 23:46, "Father, into your hands I commit my spirit!"
We now, then, have one last verse to address. We know with certainty that Jesus went to Heaven after He died, but what of His proclaiming to the spirits in prison? Who are these spirits, and where are they imprisoned? 1 Peter 3:20 says that "they formerly did not obey, when God's patience waited in the days of Noah, while the ark was being prepared." Now, Peter also spoke of Noah and spirits in 2 Peter 2, where he says, "For if God did not spare angels when they sinned, but cast them into hell [Gk. Tartarus] and committed them to chains of gloomy darkness to be kept until the judgment…" Since angels are spirit beings, and we see the same prison imagery and Noah reference here, it seems that the spirits in prison were angels who sinned and were locked in Tartarus. So, where is Tartarus? The New Testament, unfortunately, never mentions Tartarus anywhere else. However, Greek myth helps here. In Greek mythology, the underworld (ruled by the god Hades) had two compartments. There were the pleasant Elysian Fields for the upright and the torturous Tartarus for the wicked. Since Peter often alluded to extrabiblical sources to make points, it seems that Tartarus was being used to parallel the Christian concept of the abode of the wicked dead, that is, Hell. So the word appears equivalent to Gehenna, which Jesus used very often to refer to Hell. So, apparently, some of the angels sinned and were cast into Hell (one could debate whether this relates to Satan's initial rebellion or the incident with the sons of God and daughters of men), and Jesus proclaimed to them after His death and before His resurrection. What did He proclaim to them? More likely than not, He came to tell them He won. Their rebellion and alliance with Satan had proved worthless. They would be lost to eternal torment and God would be exalted. Their leader was stripped of his power.
Also note that some have a different theory about the identity of the spirits in prison. For example, in Matthew Henry's Bible commentary, he says that Christ preached through the Holy Spirit on Enoch and Noah before the flood, and these spirits are those of those who died in the flood and are now in Hell. This is entirely possible, and it would tie well with one of the explanations of the controversial 1 Peter 4:6.
What is our conclusion? It is certain that Jesus went to Heaven immediately after He died. Beyond that, there is uncertainty. There is a possibility that He descended to Hell to proclaim victory to imprisoned fallen angels. There is also a possibility that He stayed in Heaven the whole time until the Resurrection. There is no possibility, however, that Jesus spent His three days dead being punished in Hell. God's wrath against the sins Christ atoned for was totally propitiated when Jesus said, "It is finished." He had no need to bear further punishment.