The "intermediate state" is the time between the death and the resurrection. Some have held that during this time we are unconscious or possibly even go out of existence. We do not think that this is biblical.
The biblical evidence is that our soul continues on after death and that we remain conscious in the intermediate state while awaiting our final destiny of resurrected existence in the new heavens and new earth.
First, Paul spoke of having the desire "to depart and be with Christ, for that is very much better" (Philippians 1:23). Notice first of all that Paul speaks of death as a departure (from the body) not into temporary nothingness or unconsciousness but to be with Christ. If we are with Christ once we have died, then we continue existing. Second, notice that Paul speaks of this state as "very much better" than the present state. It would be hard to say such a thing of a state of complete unconsciousness. Particularly when we consider that Paul's passion was to know Christ, it would seem that the reason the state beyond death is better than this present life is because we are with Christ and know it. If we were suddenly unconscious at death until the resurrection, wouldn't it be better to remain in this life because at least then we would have conscious fellowship with Christ?
Second, Paul also said that "while we are at home in the body we are absent from the Lord" and that therefore he would "prefer rather to be absent from the body and to be at home with the Lord" (2 Corinthians 5:6-8). First, it is significant that he speaks of the possibility of being absent from the body. This implies that we indeed do have souls which continue existing after the body dies. Second, notice again that he speaks of this state as his preference, which indicates (as in Philippians 1:23) that we not only continue existing between death and the resurrection, but that we are aware of our existence.
Third, even though the thief on the cross has been used to prove about every point in Christian theology, his case is still relevant here: "And He said to him, 'Truly I say to you, today you shall be with Me in Paradise'" (Luke 23:43). The Jehovah's Witness's New World Translation punctuates Jesus words as "Truly I say to you today, you shall be with Me in Paradise," giving the impression that "today" refers simply to the time of Jesus' statement. But the context demands that the "today" refer to when the thief on the cross would be with Jesus in paradise, because Jesus is responding to his request in the previous verse: "Jesus, remember me when you come in your kingdom!" The response, "Today you shall be with Me in paradise" can in this context only be taken to mean, "Not only will I remember you when I come in my kingdom, but already today you shall be with me in heaven."
Fourth, Revelation 6:9 speaks of John seeing underneath the altar "the souls of those who had been slain because of the word of God." These individuals are surely not in a state of soul sleep because in the next verse they cry out "How long, O Lord."
John Piper, "What Happens When You Die? Part I: At Home With the Lord"
Wayne Grudem, Systematic Theology, chapter 41, "Death and the Intermediate State"