I'll do my best to explain the differences as I understand them. However, you should know that there are some differences of opinion on these items since the Bible only gives us a glimpse into some of these things.

SHEOL AND HADES - The Old Testament Hebrew word sheol and the New Testament Greek word hades mean the same thing. In the KJV Bible sheol is translated as “hell” thirty-one times, as “the grave” thirty-one times, and as “the pit” three times. In the NIV Bible it is usually translated as “the grave,” but sometimes as “death,” “the depths,” or “the realm of death.” Of course other Bible translations offer additional alternatives complicating the issue of clarity.

It’s very important to note that when the words hades or sheol are translated as “hell” they are not a reference to the “fire and brimstone” hell we often think of that is referred to in Rev. 21:8 (see the discussion on “gehenna” below). When hades or sheol are translated as “hell” they have a different meaning. In fact, they typically have one of two literal meanings (Note: the words could also be used in a figurative sense but we’re are not going to look at that here for the sake of time): (1) a literal grave; or (2) a temporary or intermediate holding place for departed souls.

(1) A literal grave: I understand that the word "hell" is actually an old English word. Several hundred years ago when the KJV of the Bible was translated to English, the people of England commonly talked of "putting their potatoes in hell for the winter” to preserve them. Hell simply meant a hole in the ground that was covered up. What some might call a grave. Over time the word began to take on additional meaning and come to mean what we typically think of today when we use the term hell. So sometimes the use of hades or sheol simply means nothing more than a literal grave (a hole in the ground) as in Rev. 20:13 mentioned above.

(2) A temporary or intermediate holding place for departed souls: A second meaning of hades or sheol relates to a common belief is it was a temporary or intermediate holding place for people who had died prior to them going to either heaven or hell. It represents a place that is different and distinct from the hell of Rev. 21:8. We’ll discuss this further below.

Although the word “hell” found in the Bible can be a reference to either the real “fire and brimstone” hell of Rev. 21:8 or to hades/sheol (a temporary or intermediate holding place for the dead) - it should be noted that hades/sheol never refer to the real “hell” of Rev. 21:8. To say it another way, wherever hades/sheol are used in the Bible (apart from figurative references) it is either referring to the temporary or intermediate holding place for the dead or simply to a grave (a hole in the ground), but never to the hell of Rev. 21:8 even though the word(s) are sometimes translated as “hell.”

THE GRAVE - The “grave” can simply mean a hole in the ground or it could be a reference to the temporary holding place called sheol or hades depending on the context in which it is used. It can also be used figuratively.

THE PIT OR ABADDON - The use in the Bible of the phrases the pit (other than “the bottomless pit” discussed below), or abbadon, or even the “place of destruction” are believed to be different ways of referring to sheol or hades, or used figuratively express pain, suffering, or loss, or to simply refer to death or destruction. It’s not likely that these are actual “places” separate and distinct from hades, or sheol, or the abyss for example. Some believe Abaddon could also refer to a person such as Satan, the antichrist, or some other powerful demonic angel. The context must be considered to know the more precise meaning of these words when found in Bible translations.

THE ABYSS (THE DEEP, OR THE BOTTOMLESS PIT) - It is derived from the Greek word “abussos” which means (1) literally, "very deep," "bottomless"; (2) figuratively, "unfathomable," "boundless." The word "abyss" is not actually used in the KJV Bible. It is found, however in varying degrees, in the NIV, ESV and other Bible translations - usually to transliterate the Greek word “abussos.”

In the KJV Bible the word abussos is translated as "the deep" in two passages (Lk. 8:31; Rom. 10:7); or the "the bottomless pit" in Revelation (Rev 9:1,2,11; 11:7; 17:8; 20:1,3).

In the Septuagint (Greek translation of the Old Testament) “abussos” is how Jewish translators rendered the Old Testament Hebrew word “tehom.” The Septuagint never uses “abussos” as a rendering of the Hebrew word sheol (which is translated into Greek as hades and means “place of the dead”). Therefore it is believed that “tehom” (and by default its Greek counterpart abussos) never meant an "abode or place of the dead," which it has unfortunately been translated as in a few instances. For example, in the New Living Bible translation of Romans 10:7 the word abussos is rendered as the “place of the dead,” and the Complete Jewish Bible translation of that same passage renders abussos as “Sheol,” which has created some confusion. But the weight of evidence seems to indicate that the Greek abussos and its Hebrew equivalent tehom refer to something very different than the Greek hades and its Hebrew equivalent sheol.

The most likely meaning of abyss is a temporary “abode of demons” (not of departed humans). It’s a place to hold at least some of them until they are eventually cast into the lake of fire after the millennial reign of Christ. It is probably connected to a place described 2 Peter 2:4 by the Greek word “tartaros” - see discussion below.

Support for this idea is found in Luke 8:31 which uses the word abussos in reference to demons not humans. In this passage the KJV renders abussos as "into the deep" (Weymouth and The Twentieth Century New Testament = "into the bottomless pit"). In this passage the demons do not want Jesus to send them to the abyss, which is implied is a holding place of punishment until the end times. In this same account in the Gospel of Mark 5:10 the Greek word “chora” is used. The KJV translated chora as “out of the country,” meaning Jesus would send them away. Chora can mean “country” but it also means “a space lying between two places or limits” (Strong’s Ex. Concordance). A clearer understanding of Mark 5:10 would be that the demons feared Jesus would send them to a place that is “between” this world and their final holding place “the lake of fire.” This would be consistent with Luke’s use of the word abussos or “abyss.”

Also Rev. 9:11 speaks of an angel king of the abyss named Abaddon or Applyon who is released at the sounding of the fifth trumpet when the “abyss” or “bottomless pit” is opened during the tribulation period. This supports the idea that the abyss is an abode of evil spirits (not humans), but not their place of final punishment; it is therefore to be distinguished from the "lake of fire and brimstone" where the beast and the false prophet are, and into which the Devil is to be finally cast (Rev 19:20; 20:10). To summarize, the abyss is a temporary holding place for demonic spirits (not departed humans) to secure these spirits and cut them off from this world until they can be thrown into the “lake of fire” after the millennial reign of Christ.

TARTAROS - Tartaros is a Greek word, which has also been translated into English as "hell" creating even more confusion about what hell is. It occurs only once in the New Testament in II Peter 2:4. This passage does not refer to humans but to some kind of restrained condition of fallen angels. It means "darkness of the material universe," or "dark abyss," or "prison." But again it has nothing to do with people. It seems to refer to a temporary holding place or condition for fallen angelic beings and/or demonic spirits. It is possible just another reference to the “abyss” mentioned earlier.

GEHENNA, THE LAKE OF FIRE, AND HELL - Gehenna, the lake of fire and hell (the real fire and brimstone hell of Rev. 21:8) are all the same thing. Again what makes this confusing is that translators have used the word hell to refer to both the Rev. 21:8 gehenna hell and to the very different place of hades/sheol (the temporary holding place of the dead). Sometimes the only way to tell the difference is to use a concordance to look up whether the actual word used was gehenna (Rev. 21:8 “hell”) or hades (temporary holding place of the dead).

Gehenna is a Greek word often translated as “hell” in the New Testament. The word is actually derived from the name of the narrow, rocky Valley of Hinnom, which was a garbage dump on the south side of Jerusalem. It was the place where garbage was constantly being burned. Trash, filth, and even dead bodies of animals and despised criminals were thrown into the fires of gehenna, or the Valley of Hinnom.

Psalm 23 refers to “valley of the shadow of death.” The Psalm is believed to have been written by King David reflecting on a time when he had to flee Jerusalem in fear for his life when his son tried to forcefully take the throne. Some believe that he fled the city by passing through the Valley of Hinnom and when writing the Psalm called this place the “valley of the shadow of death” which would have had multiple meanings. One meaning referring to activity of garbage and dead things being burned there, one meaning referring to his precarious situation in fleeing for his life through that valley, and one meaning being prophetic and referring to this place as picture of real and future gehenna hell consistent with how Jesus described hell.

Ordinarily, everything thrown into this valley was destroyed by fire—completely burned up. Therefore, Christ used gehenna to help us picture the terrible fate of unrepentant sinners. Gehenna is the final place of punishment. It is the “fire and brimstone” hell of the Bible.

Gehenna is the hell of Revelation 21:8 (NIV), “But the cowardly, the unbelieving, the vile, the murderers, the sexually immoral, those who practice magic arts, the idolaters and all liars—their place will be in the fiery lake of burning sulfur. This is the second death."

Jesus used the word gehenna in Mark 9:43,45,47 Luke 12:5 to teach that the unrighteous will be punished by being put into “hell,” (gehenna fire) which He describes as a fire that will not be quenched (see also Jeremiah 17:27).

The idea of gehenna is also used in Matthew 7:15-19; 13:30; 23:33; Matthew 8:12; 22:13; 25:30; Matthew 13:42, 50; and Hebrews 10:26-27. All who stubbornly refuse to repent and persist in breaking God's commandments ultimately find themselves in a lake of "fire and brimstone." And a very large fire would have the appearance of a fiery lake, hence its description. The fate of the wicked is gehenna fire.

So gehenna, the lake of fire, and what most people would typically think of as “hell” today are all the same. The Bible makes it clear that the lake of fire or hell was not intended for people but for the devil and his angels (Matthew 25:41).

THE SECOND DEATH - The “second death” is more of a condition than a place. The Bible calls final separation from God “the second death.” The second death is what happens to someone who is cast into gehenna, or the lake of fire, or hell (they are all the same thing - just different names).

In Revelation the lake of fire is described as the second death (Revelation 20:14). Jesus also identified gehenna as a second death when He warned: “Do not be afraid of those who kill the body but cannot kill the soul. Rather, be afraid of the One who can destroy both soul and body in hell [Gehenna]” (Matthew 10:28; see also Luke 12:4,5). It clearly refers to another death after the physical death of the body and that is different than physical death. As physical death is separation from the body and from the environment of this life, so the second death is a final and eternal separation from God and from the life to be enjoyed in the new creation.

Those who will experience the second death include all who take the mark of the beast (Revelation 14:9–11). They will be tormented with fire and brimstone in the presence of the angels and Christ. There is no promise of rehabilitation or restoration once the final judgment is pronounced. Gehenna is forever!