Concise Bible Dictionary:
The state which the bodies of the dead in Christ (now going to corruption in their graves) will have at the resurrection. When this takes place, and the mortal shall have put on immortality, death will be swallowed up in victory! (1 Cor. 15:42-54). In Romans 2:7 and 2 Timothy 1:10 the word is the same as the above, and should be translated “incorruption” instead of “immortality.”
From Anstey’s Doctrinal Definitions:
This refers to the deathless state into which believers will be changed when they are glorified. For the New and Old Testament saints, this will occur at the time of the Rapture (1 Cor. 15:51-56; Heb. 11:40). For the martyred portion of the believing Jewish remnant, it will be at the end of the Great Tribulation (Rev. 14:13). Old Testament saints didn’t know much about life after death, but the truth concerning it has now been brought to light through the gospel. We now know that there is “life” for the soul, and “incorruptibility” for the body (2 Tim. 1:10).
As mentioned, at the moment of the Rapture, the Lord will effect a change in the saints who have died, which Paul states as: “this corruptible must put on incorruptibility” (1 Cor. 15:53a). The Lord will also effect a change to the living saints which Paul states as: “this mortal must put on immortality” (1 Cor. 15:53b). This change will be both moral and physical. Their souls and spirits will be rid of the fallen sin-natures, and thus they will be made like Christ morally (1 John 3:2). Also, their bodies will be made like Christ’s body physically (Phil. 3:21).
The souls of all men are immortal—regardless of whether a person is saved or lost. Genesis 2:7 says, “And the LORD God formed man of the dust of the ground, and breathed into his nostrils the breath of life; and man became a living soul.” This shows that the spirits and souls of men are “living,” not dying. All such will live on forever, either in the enjoyment of the blessing of God or under damnation. Even after a person dies and his body is buried, his spirit and his soul live on. In connection with this, the Lord said, “All [the dead] live unto Him” (Luke 20:38). While man’s soul is immortal, his body is “mortal”—subject to death (Job 4:17; Rom. 6:12; 8:11; 1 Cor. 15:53-54; 2 Cor. 4:11). The good news is that there will come a time for the believer when “mortality” will be “swallowed up of life” (2 Cor. 5:4).