What exactly does the doctrine of perseverance teach?

The historic, biblical doctrine of the perseverance of the saints does not teach that God's children are above committing sin, above deception, or above faltering in their faith, nor does it teach that God's children cannot be involved in such errors at the point of their death. Rather, perseverance means that notwithstanding these things, God's born-again children forever possess a spiritual nature that disposes them against sin and against the lies of Satan and against unbelief (1 John 3:7-9; Mark 13:22; John 3:36). God's children may act contrary to their spiritual nature when under the temptations of the flesh. This is illustrated by the experience of every saint in the Bible; nonetheless, the profound change effected in the heart by the new birth cannot be undone. While during a time of failure such as Peter denying the Lord, one of the Lord’s children may not display evidence of grace to other men’s satisfaction, the Lord continually knows the heart, and faithfully and effectively chastens every one of His children (Hebrews 12:6; Revelation 3:19). We should be "zealous therefore, and repent.”