In the old testament, when people died they were said to descend to Sheol. It wasn't particularly spiritualized as much as it was just an understanding that dead people go to the grave and the grave swallows them up. There was an understanding that the spirit is not annihilated, but not a completely articulated theology of some division into those in punishment and those in paradise.
In Greek Sheol is Hades, and in English, Hell. Within the New Testament, we are confirmed that the thief on the cross with Jesus would be with him in paradise "today", i.e. not on the last day in the resurrection. Paul also assures us that to be absent from the body is to be present with the Lord, presumable, if you are saved.
So it's relatively clear that upon the resurrection there will be a resurrection unto life with a new heavens, a new earth, and a new Jerusalem. There will be a resurrection unto death with a lake of fire and the smoke of their torment rises forever. What is not completely clear to me is that presently dead individuals immediately enter into a state of torment while awaiting the resurrection.
The go-to proof for this is the parable of the rich man and Lazarus. On the one hand, it seems to discussing Sheol, not the resurrection. On the other hand, it is after all a parable and might not be a good place to draw a complete theology of the afterlife. In addition, it would introduce a paradox. To be "absent from the body is to be present with the Lord", but the rich man apparently has a tongue which wants water, and his body is in anguish. He seems to have a body. This isn't a problem if it is indeed a parable, but it also would take away the legitimacy of using this parable to tell us something about the unbelieving dead, right now.
Jesus also spoke about "Hell", but as Gehenna, not Sheol/Hades. It is clear that this is a place that "soul and body" are destroyed in flames. The alternate of "entering life" is something that Jesus seems to say is something that it would better to "enter lame". Again, problematic. I thought the body is restored and glorified in life. Many people point to this passage as an example of hyperbole. Some (universalists) postulate that in these passages, Jesus is referring to the literal Gehenna where idolaters and idols perished with an allusion to the siege of Jerusalem by the Romans, as if to say that the immoral would literally die in this manner. There might be an implication this is true, but even so, it would be true in an apocalyptic sense as well, in which human events with spiritual meanings are said to take on a cosmic dimension. Thus it would make sense that Jesus is actually talking about the second death and might be foreshadowing their first death in the siege.
But so it's all quite confusing. What happens when the unbeliever dies before the resurrection? Do they suffer a bodily torment while absent from the body and prior to the verdict of their judgment? What then is even accomplished in the judgment? In Revelation, the martyrs seem eager for their blood to be avenged. If there is torment in the intermediate state, what's the difference between the lake of fire and torment after death?