Definition of Hell:

1. In the Old Testament, Hell is Sheol. 
2. In the New Testament, Hell is Gehenna or Hades.

Note: There has been a great volume of speculation regarding hell, and this goes back hundreds of years.  Problems arise whenever anyone goes beyond Scripture, that is, when anyone adds to Scripture or takes from Scripture.  A theology has been developed that goes beyond the Bible on the one hand by saying such things as "there is no hell," "there is a hell, but hell is not all that bad," "God would never correct anyone through punishment," or "all ways lead to God."  I'm sure that there are other examples of this extreme view.  On the other hand, a robust theology has developed beyond the Bible that makes the Scripture conflict with itself.  This is a problem if we are to believe the entire Bible.  This extreme dogma causes some theologians to find extra-biblical reasons to rationalize away the Scriptures that plainly say such things as, "As in Adam all die, so in Christ shall all be made alive."  Some people do not, as a result, believe in the restitution of all things, which is also a plain teaching of Scripture just as Hell is a plain teaching of Scripture.  Sometimes, the best answer is, "We believe it all, but God has not yet told us how He is going to do this."

Calvinism & Arminianism: Man is totally depraved.
Calvinism: Jesus only atoned for the sins of the elect.
Arminianism: Jesus atoned for the sins of all people.
Calvinism: Election is unconditional. 
Arminianism: Election is conditional.
Calvinism: God’s grace is irresistible. There is no human free will.
Arminianism: People can resist the grace of God.
Calvinism: A person who is elected by God cannot permanently deny Christ. (Eternal Security doctrine)
Arminianism: A person can fall away from Christ.

There are many variations within these systems, but both systems try to explain Divine revelation using the human mind. Both agree that man is totally depraved, so both are self-refuting. Some variations of Arminianism deny the total depravity of mankind. In effect, they maintain that human reason is valid. Human reason is valid so long as it doesn't violate the laws of logic. Logic requires a true premise. The only sources for premises are Divine revelation and making things up. Logic requires that the conclusion adds nothing to the premises. For this reason, any assumptions are irrational. All that we can know is what God has revealed. Assumptions add to His Words or subtract from His Words, whether or not we are aware that we're making assumptions.

Calvinist theology tends to assert that Hell exists, but God has foreordained some people to Hell. They don't know who they are. They can't do anything about it. They are just going to Hell. There's nothing to be done for it, since they have no choice in the matter. God decided ahead of time who was going to go to Heaven.

Scripture only says that God set some ahead of time who would be conformed to the Image of Christ--but this setting was based on foreknowledge. Just as a good project manager chooses qualified people to perform certain tasks, God chose those who would make the choices needed. 

Much of this centers on the definition of "for ever". A label of "universalist" is assigned to anyone who doesn't believe in Hell than never ends. The term, "universalist," makes no distinction between those who believe that Hell doesn't exist and those who believe in Hell as taught by the Bible, but don't agree with the translation of "for ever." That makes "universalist" a slur and a meaningless word. Slurs are used to control people through intimidation.

07585 לואשׁ sh@'owl sheh-ole' or לאשׁ sh@ol sheh-ole'
from 07592; n f; {See TWOT on 2303 @@ "2303c"}
AV-grave 31, hell 31, pit 3; 65
1) sheol, underworld, grave, hell, pit
1a) the underworld
1b) Sheol-the OT designation for the abode of the dead
1b1) place of no return
1b2) without praise of God
1b3) wicked sent there for punishment
1b4) righteous not abandoned to it
1b5) of the place of exile (fig)
1b6) of extreme degradation in sin

1067 γεεννα geenna gheh'-en-nah
of Hebrew origin ם "־איק, 01516 and 02011; TDNT-1:657,113; n f
AV-hell 9, hell fire + 3588 + 4442 3; 12
1) Hell is the place of the future punishment call "Gehenna" or "Gehenna of fire". This was originally the valley of Hinnom, south of Jerusalem, where the filth and dead animals of the city were cast out and burned; a fit symbol of the wicked and their future destruction.

86 Aδης hades hah'-dace
from 1 (as negative particle) and 1492; TDNT-1:146,22; n pr loc
AV-hell 10, grave 1; 11
1) name Hades or Pluto, the god of the lower regions
2) Orcus, the nether world, the realm of the dead
3) later use of this word: the grave, death, hell