I will give a scripture reference or two for each one, the actual definition from a reliable dictionary, and then offer a few comments.

SHEOL: Genesis 37:35 reads, “Then all his (Joseph’s) sons and all his daughters arose to comfort him (Jacob), but he refused to be comforted. And he said, ‘Surely I will go down to Sheol in mourning for my son.’ So his father wept for him” (NASB). Here is the definition for SHEOL (in this verse) from the Hebrew Dictionary in Strong’s Exhaustive Concordance: “HADES or the world of the dead.” Some translations use the word “grave” in this verse, but that it clearly wrong, for “Sheol” is actually “the place of departed spirits/souls.” In this verse Jacob was led to believe that his son, Joseph, had died and though he didn’t know where his body was buried, he informs his sons and daughters that he would mourn for him UNTIL he died and was reunited with Joseph in “the world of the dead.” So, when a person dies, their “body goes into a grave” but their spirit/soul goes to “the world of departed spirits/souls.” Another verse that speaks of “sheol” that illustrates this truth in Psalm 16:10, “For You will not abandon my soul to Sheol; nor will You allow Your Holy One to undergo decay.” This verse is “Messianic”; that is, it speaks of the Lord Jesus Christ (see Acts 2:24-31 which confirms this). The Lord Jesus is the speaker is this wonderful verse and it He is basically saying to His heavenly Father, “You will not leave my soul in the place of the dead, nor will You allow My body to decay in the grave.” This precious verse teaches us the truth that after Jesus died His body was buried, His spirit went to the place of departed spirits (which in His case was heaven…see Luke 23:46), and (it is implied) that three days later His body was resurrected and reunited to His spirit. If you read the gospel accounts of Jesus’ death and resurrection you will see that this prophecy was indeed fulfilled! It should be noted here that Sheol (and Hades, as we shall see) is not only a PLACE, but a CONDITION. It is the “condition of the spirit being separated from the body.”

HADES: Luke 16:22-23 read, “Now the poor man died and was carried away by the angels to Abraham’s bosom; and the rich man also died and was buried. In Hades he lifted up his eyes, being in torment, and saw Abraham far away and Lazarus in his bosom” (NASB). The definition for HADES in this verse from the Greek Dictionary in Strong’s Concordance is, “The world of the dead.” It has the exact same meaning as the Hebrew word “Sheol” for they are, in truth, one and the same thing. In the passage we just quoted the rich man’s body was buried in a grave and his spirt/soul went to the place of the dead. In his case, “the place of the dead” was a “place of torment,” whereas the poor man’s (Lazarus) spirit/soul went to heaven to be with Abraham and the departed spirits of other believers and for them it was a “place of comfort.” I had referred you earlier to Acts 2:24-31 to see that Psalm 16:10 had its fulfillment in Christ’s death and resurrection. In verse 27 we have the Apostle Peter’s quotation from Psalm 16:10, “Because You will not abandon my soul to HADES, nor allow Your Holy One to undergo decay.” Notice how the words are identical to Psalm 16:10 except for one word; Peter uses the word HADES instead of SHEOL, for he HADES is the Greek word for SHEOL. Again, their meaning is exactly the same.

TARTARUS: 2nd Peter 2:4 states, “For if God spared not the angels who had sinned, but having cast them down to the DEEPEST PIT OF GLOOM has delivered them to chains of darkness to be kept for judgment” (DARBY version). The actual word TARTARUS is not found here, yet the word “tartarus” in the Greek means, “a horrible pit of torment,” so the words “the deepest pit of gloom” are the same as “tartarus.” This is NOT the same as HADES, where in the case of lost souls they too suffer torment in “the place of departed spirits/souls.” Tartarus is a special “pit of torment” reserved for a special class of angels who rebelled against God and who are being kept in this pit until the final judgment. We believe Satan (and his angels) will also join them in this pit, as we see in Revelation 20:2-3, “And he laid hold of the dragon…who is the devil and Satan, and bound him for a thousand years; and he threw him into the ABYSS.” The “abyss” is the same as the “deepest pit of gloom,” which is “tartarus.”  (359.3)  (DO)