The Bible mentions three different types of hell.  What are they?

The Grave – sheol and hades

In the Old Testament there is just one Hebrew word that is translated as hell, and that is sheol.  It means grave, hell, pit, or underworld (where the dead reside).  Here are a couple of examples.

Psa 9:17  The wicked shall be turned into hell [sheol], And all the nations that forget God.

Psa 55:15  Let death seize them; Let them go down alive into hell [sheol], For wickedness is in their dwellings and among them.

The equivalent word to sheol in New Testament is the Greek word hades.  It also means grave, pit, world of the dead, or hell.  Here are some examples.

Luk 10:15  And you, Capernaum, who are exalted to heaven, will be brought down to Hades [hades].

Act 2:31  he, foreseeing this, spoke concerning the resurrection of the Christ, that His soul was not left in Hades [hades], nor did His flesh see corruption.

These are quoting from the NKJV which translates hades as “Hades”, but many other Bible translations translate hades as “hell”.

Incarceration – tartaros

The second Greek word translated as “hell” is tartaros.  It means “incarceration, captivity, or a dark abyss” and is found only once in the New Testament.

2Pe 2:4  For if God did not spare the angels who sinned, but cast them down to hell [tartaros] and delivered them into chains of darkness, to be reserved for judgment;

This verse describes the imprisoning of the angels on earth as their “place of restraint” or “prison” after their rebellion.

Rubbish tip – gehenna

The third Greek word translated hell in the New Testament is gehenna.  It comes from Hebrew and literally means the Valley of Hinnom, a valley just south of Jerusalem where people tipped their rubbish, filth, and animal carcasses for burning.  Gehenna was a rubbish tip, and Gehenna fire was the fire lit in the rubbish tip to burn the rubbish.  Here are some examples.

Mat 5:22  But I say to you that whoever is angry with his brother without a cause shall be in danger of the judgment. And whoever says to his brother, ‘Raca!’ shall be in danger of the council. But whoever says, ‘You fool!’ shall be in danger of hell [gehenna] fire.

Mat 10:28  And do not fear those who kill the body but cannot kill the soul. But rather fear Him who is able to destroy both soul and body in hell [gehenna].

Mat 18:9  And if your eye causes you to sin, pluck it out and cast it from you. It is better for you to enter into life with one eye, rather than having two eyes, to be cast into hell [gehenna] fire.

Mat 23:33  Serpents, brood of vipers! How can you escape the condemnation of hell [gehenna]?

Jas 3:6  And the tongue is a fire, a world of iniquity. The tongue is so set among our members that it defiles the whole body, and sets on fire the course of nature; and it is set on fire by hell [gehenna].

Often the word Gehenna is used with fire.  This is a reference to the “lake of fire” described in Revelation.

Rev 19:20  Then the beast was captured, and with him the false prophet who worked signs in his presence, by which he deceived those who received the mark of the beast and those who worshiped his image. These two were cast alive into the lake of fire burning with brimstone.

Rev 20:10  The devil, who deceived them, was cast into the lake of fire and brimstone where the beast and the false prophet are. And they will be tormented day and night forever and ever.

Rev 20:14  Then Death and Hades were cast into the lake of fire. This is the second death.
Rev 20:15  And anyone not found written in the Book of Life was cast into the lake of fire.

Rev 21:8  But the cowardly, unbelieving, abominable, murderers, sexually immoral, sorcerers, idolaters, and all liars shall have their part in the lake which burns with fire and brimstone, which is the second death.”

This lake of fire is the second death, Rev. 20:14.  The first death is what all humans experience when we die.  In fact, it is appointed (part of God’s plan) that we all die once.

Heb 9:27  And as it is appointed for men to die once, but after this the judgment,

The first death is temporary.  Some rise (at the first resurrection) from this first death to everlasting life, Rev 20:4-5. 

Rev 20:4  And I saw thrones, and they sat on them, and judgment was committed to them. Then I saw the souls of those who had been beheaded for their witness to Jesus and for the word of God, who had not worshiped the beast or his image, and had not received his mark on their foreheads or on their hands. And they lived and reigned with Christ for a thousand years.
Rev 20:5  But the rest of the dead did not live again until the thousand years were finished. This is the first resurrection.

And some rise (at the second resurrection) from this first death to judging, Rev 20:12-15. 

Rev 20:12  And I saw the dead, small and great, standing before God, and books were opened. And another book was opened, which is the Book of Life. And the dead were judged according to their works, by the things which were written in the books.
Rev 20:13  The sea gave up the dead who were in it, and Death and Hades delivered up the dead who were in them. And they were judged, each one according to his works.
Rev 20:14  Then Death and Hades were cast into the lake of fire. This is the second death.
Rev 20:15  And anyone not found written in the Book of Life was cast into the lake of fire.

The second death, Rev 20:14, is what those who enter this lake experience; they suffer permanent death.   They suffer complete destruction; a final punishment that is everlasting, eternal, permanent!  It is not the punishing that is everlasting but rather the punishment that is everlasting.