Hades and Sheol were the general-use afterlife locations for the Greeks and Jews, respectively. They were extremely similar in description as dark, gloomy, colorless places in which the spirits of the dead sort of moldered on, forever. (Jewish tradition has one theme that indicates that death is, itself, oblivion. Sheol represents a separate tradition, but whether this similarity to Hades was the result of a common belief that permeated many cultures at the east end of the Mediterranean, centuries before the common era, or whether the notion of a Hades-like place was adopted by the Jewish people in the post-Alexandrine expansion of Hellenistic beliefs, I have never seen addressed.)
Gehinnon was a valley to the southeast of Jerusalem where the people of Jerusalem tended to throw their trash. Jeremiah described the bodies of sinners being thrown on to the constant, smoldering fires that burned there, providing a metaphor for the destiny of evil people.
Tartarus was a place in Greek mythology where deposed gods and Titans were imprisoned. Elysium was a corresponding place in which demi-gods, (born to gods and humans), and heroes, (who tended to be demi-gods, anyway), enjoyed happiness after death. Hades remaining the location of the spirits of normal humans.
Eventually, the notions of Tartarus and Elysium were expanded so that good persons would end up in the Elysian Fields while evil doers would be punished in Tartarus.
In the period from around 250 B.C.E. to around 150 C.E., a literary style developed within Judaism and later borrowed by early Christianity that we now call apocalyptic, from the Greek word for revelation. That is the style in which the Revelation of John, (or Apocalypse of John), was written. It is characterized by wildly imaginative depictions of events, typically future events that are being revealed, (hence the name). Few of those works were eventually accepted as Scripture by either the Jews or the Christians. (Revelation and several passages in the Prophet Daniel being the notable exceptions.)
When Christians began identifying the books that would become their canonical scripture, most of the apocalyptic books were set aside and are known as apocryphal, (“set aside” or “hidden”) works. However, while they did not get accepted as scripture, they did influence the beliefs of many people. It was among these writings that the idea developed, (much as the similar idea had developed among the Greeks), that all people would continue in an afterlife and that they would be judged, resulting in their placement in either a place of joy or a place of punishment. In those works, Gehenna and Tartarus figure as places of tormenting punishment, Gehenna, particularly, is described as unending fire.
As Christianity developed, it brought along many of the ideas from both the Greek and Judaic traditions. Since the Apocalyptic literature is not read each Sunday in services, people who examine only scripture are not always aware of the trends that led to the current formations of ideas about heaven and hell in Christian thought.
As the literary language of the first two centuries was Greek, the authors of the New Testament tended to write in Greek. However, several of them had grown up in the Jewish tradition. Hence, they tended to borrow words to describe these places of the afterlife without rigorously setting forth definitions for them. Thus, we get numerous references to Hades, Sheol, and Gehenna in the New Testament, and one reference to Tartarus. The words are not always used in a rigidly defined manner, for example the parable of Lazarus and the rich man, in Luke 16:19 - 31, uses the word Hades, even though the description is one of torment.
It should be noted that while Gehenna is described as a place of torment with unending fire, the torment, itself is not described as unending with the exception of one ambiguous mention in Matthew.
Following on Jewish thought of the first century, punishment in the afterlife is described by some as a period of torment, (one year in various Jewish sources), followed by oblivion. The one ambiguous comment by Matthew, along with similar remarks in the apocryphal First Book of Enoch, Second Book of Esdras, and The Assumption of Moses seem to have captured the imagination of early Christians, leading to the concept of Hell, itself, being a place of eternal suffering.
Attempting to rigidly classify “places” in the New Testament based on their names is pointless. We do not really know which author followed a religious (or secular) tradition that prompted him to use one name or another. And, as we see in Luke 16, they could even use names that were inappropriate to the message they were delivering.