What is hell?

I believe the bible teaches that there are two hells. One is in Sheol which is inside the earth and is a holding tank for the second hell which is the lake of fire. Sheol is described in Luke 16 so I'll skip to the lake of fire:

Revelation 21:8
But as for the cowardly, the faithless, the detestable, as for murderers, the sexually immoral, sorcerers, idolaters, and all liars, their portion will be in the lake that burns with fire and sulfur, which is the second death.”

The Greek word rendered faithless here is "apistos" (Strong's greek word 571).

http://biblehub.com/greek/571.htm

571 ápistos (from 1 /A "not" and 4103 /pistós, "faithful," see there) – properly, not faithful because unpersuaded, i.e. not convinced (persuaded by God). 571 /ápistos("faithless, unpersuaded") does not always refer to the unconverted – see Jn 20:27. 571 (ápistos) describes someone who rejects or refuses God's inbirthings of faith (note the root, faith, 4102 /pístis).

This is the same word used in John 20:27 when Jesus described Thomas, someone who was his disciple!

John 20:27
Then he said to Thomas, “Put your finger here, and see my hands; and put out your hand, and place it in my side. Do not disbelieve, but believe.”

It should be noted that the list given in Revelation 21:8 is similar to the ones given in 1 Cor 6:9-10, Gal 5:19-21, Eph 5:3-5 which describes conditions for not inheriting the kingdom of God. Is it possible that not inheriting the kingdom of God implies going to hell? Jesus seems to confirm this idea by juxtaposing hell versus inheriting the kingdom of God in Mark 9:

Mark 9:47-48
And if your eye causes you to sin, tear it out. It is better for you to enter the kingdom of God with one eye than with two eyes to be thrown into hell, ‘where their worm does not die and the fire is not quenched.’

It's also interesting Revelation 21:8 mentions the cowardly. This should remind us of the ten spies, who despite being  handpicked from their respective tribes,  disbelieved in God's promises when they saw thirty foot tall giants in Canaan. Even though they were Israelites who "believed" in Jehovah and were children of the promise, they were severely judged by God due to their cowardice.

In the OT, the judgment of the lake of fire is also alluded to which is different from the Jewish understanding of Sheol (the place you go after you die):

Daniel 12:2
And many of those who sleep in the dust of the earth shall awake, some to everlasting life, and some to shame and everlasting contempt.

So putting these verses together: The lake of fire is the second death where one experiences everlasting shame and contempt where the worm does not die and the fire is not quenched.