Tartarus is only used once in all of scripture and is specific to the angels who sinned. Not sure where Tartarus is, but it could be connected with water.
For if God did not spare the angels who sinned, but cast them down to Tartarus and delivered them into chains of darkness, to be reserved for judgment...
-2 Peter 2:4
Then the sixth angel sounded: And I heard a voice from the four horns of the golden altar which is before God, saying to the sixth angel who had the trumpet, “Release the four angels who are bound at the great river Euphrates.” So the four angels, who had been prepared for the hour and day and month and year, were released to kill a third of mankind.
Another word often unfortunately translated as "hell" (like Hades and Tartarus) is Gehenna. Gehenna is a specific place in the land (the Valley of Hinnom or the Valley of Slaughter - Jer 19:6). It is where a great future fire will burn (literally) and "carcasses" will be thrown in it. As with "the gates of Hades," the Lord is quoting Isaiah and not introducing a new idea when he speaks of the place where "their worm shall not die, neither shall their fire be quenched" (Mark 9:43-48). Again, the context of Isaiah 66 is a future event, concerning the land.
And they shall go forth, and look upon the carcasses (Hebrew: פֶּגֶר "peger" dead bodies) of the men that have transgressed against me: for their worm shall not die, neither shall their fire be quenched; and they shall be an abhorring unto all flesh.
If you read the entire passage, the context is the fulfillment of the New Covenant for Israel (which is hers and still future). She will take her place as a witness to the nations (Gentiles) and her place as a Kingdom of priests (the subject of the covenants). That's a bigger topic (The New Covenant).
The gift of resurrection Life has always been a free gift, by grace, through faith, since Adam. Nothing can alter that. However, there is a judgment of service and faithfulness for all servants. There are also different families of God (Eph 3:15) which will come together only alluded to in scripture. Before that, there are several different hopes.
The Hopes of Scripture
Some are hoping for a restored Paradise, others in the restored Kingdom (the subject of the Lord's earthly ministry and the apostles to the circumcision). The kingdom was forbidden to be preached to Gentiles or outside the land. (Matthew 10:5-7; Matthew 15:24; Romans 15:8; etc.)
This ministry had to do with the earth and the land. In fact, most of scripture concerns these hopes. There is a greater hope which is the New Jerusalem which comes down from the heavens to the earth. This is the hope of the "better resurrection" (Heb 11:35), "the city... whose builder and maker is God" (Heb 11:9-11). Then there is a hope which is in the "far above the heavens" (or in heavenly places). (Eph1:3; Eph 2:6). This was hidden from BEFORE the foundation of the world (Eph 1:4). The other hopes are FROM or SINCE the foundation of the world.
Paul preached only that which was taught by Moses and the Prophets in the Acts. He testified to this in his ministry and at his trial (Acts 22:26). But in Ephesians, he reveals a Mystery (there are several Mysteries in scripture), that is, a secret, which was given to Paul alone (Eph 3:8-12). It was "hidden in God from the foundation of the ages." It has do with heavenly places and has nothing to do with the promises to Abraham (the land) or to David (the promise of a Kingdom). Note that the 12 disciples are promised to "sit on 12 thrones" in the land in the kingdom "judging the 12 tribes of Israel". Earthly callings.
Heavenly places is where Christ sits. It is the place of true Holy of Holies (the earthly one was but a shadow) where Christ is building the true heavenly temple (Eph 3:19-22). Currently, the heavenly places are corrupted with the evil principalities and powers (they're not in "hell" ; Eph 3:10; Eph 6:12).
The earthly hopes have much to do with Israel as does the New Jerusalem (which has the names of the 12 tribes on it as well as the names of the 12 apostles to the circumcision). All, of any family of God, are saved by grace through faith, and all will face a judgment for service.
For we must all appear before the judgment seat of Christ, that each one may receive the things done in the body, according to what he has done, whether good or bad.
Bondservants, obey in all things your masters according to the flesh, not with eyeservice, as men-pleasers, but in sincerity of heart, fearing God. And whatever you do, do it heartily, as to the Lord and not to men, knowing that from the Lord you will receive the reward of the inheritance; for you serve the Lord Christ. But he who does wrong will be repaid for what he has done, and there is no partiality.
Those Cast Into Outer Darkness
We first see this in Matthew 8. All we have at this point is the proclamation of the King and the Kingdom from John the Baptist and the Lord Jesus Christ. No idea or mention of his death.
And I say to you that many will come from east and west, and sit down with Abraham, Isaac, and Jacob in the kingdom of heaven. But the sons of the kingdom will be cast out into outer darkness. There will be weeping and gnashing of teeth.”
This is a judgment of "the sons of the kingdom." How many preachers point that out? The context is Israel and Gentiles. Some Gentiles will come and sit down with Abraham, but some sons (heirs) of the Kingdom will be cast out among the nations for faithlessness or poor service.
Gonna try and finish by noting the two other references. The next is in Matthew 22 regarding a "guest" from outside the land ("the highways") not having a wedding garment. We know from the Revelation (another Kingdom book) that white garments are "the righteous acts of the saints" (Rev 19:8) and is connected to the wedding of the Lamb (Rev 19:9). The overcomers are clothed in white garments (Rev 2 & 3).
The last occurrence of the casting out in Matthew is in regard to "servants."
For the kingdom of heaven is like a man traveling to a far country, who called his own servants and delivered his goods to them.... “But his lord answered and said to him, ‘You wicked and lazy servant, you knew that I reap where I have not sown, and gather where I have not scattered seed. So you ought to have deposited my money with the bankers, and at my coming I would have received back my own with interest. So take the talent from him, and give it to him who has ten talents. ‘For to everyone who has, more will be given, and he will have abundance; but from him who does not have, even what he has will be taken away. And cast the unprofitable servant into the outer darkness. There will be weeping and gnashing of teeth.’
Unbelievers are not the Lord's servants. There are two paths all believers are on (no matter the hope in view). We are either on the road to faithfulness and perfection (maturity) or on the road to unfaithful service and perdition (loss). Life is not in question, but reward.
And faithfulness includes being faithful to the calling to which we have been called (Eph 1:18; Eph 4:1). Not just "obeying the Bible." I can build an ark. I can start walking to Jerusalem. I can build a tabernacle. I can build a temple. I can slay Amorites. I can believe one of my descendants will sit on the throne in Jerusalem. I can sleep on my side for 40 days, I can start out for Nineveh, I can observe days, feasts, and ordinances. I can try to observe the Ten Commandments. I can place myself under various covenants. But God has not called us unto those things. Not everything is for all people of all ages in all hopes.
We have the highest and greatest calling in this age! We have "unsearchable riches in Christ!"
So we can be wicked in our bodies (adulteries, fornications, greed, hatreds, etc. etc. etc.) or we can go about busying ourselves trying to please God in the flesh according to another calling for another hope. Either way, I cannot lose the free gift, but I can lose reward, crowns, or the Prize of the High Calling (Phil 3:14). Note, our calling is a "high calling." It has to do with a place in the far above the heavens. This is why we shouldn't busy ourselves with the ordinances and commands regarding the land, the kingdom, and the earth (Col 2).