Does the Bible say anything about what demons *are*? I've thought all my life that demons (and evil spirits) were fallen angels, but I recently encountered the viewpoint that these demons are actually the souls of men who have already died but wouldn't go to Heaven. Hence, what does the Bible say about this?

Specifically, what biblical references speak to whether or not demons originate as human or as another species?
The concept of fallen angels is based on the following scriptures.


  MATTHEW 25:41 Then shall he say also unto them on the left hand,
  Depart from me, ye cursed, into everlasting fire, prepared for the
  devil and his angels:
  
  Luke 10:17-18 And the seventy returned again with joy, saying, Lord,
  even the devils are subject unto us through thy name. And he said unto
  them, I beheld Satan as lightning fall from heaven.
  
  Jude 1:6 (KJV) And the angels which kept not their first estate, but
  left their own habitation, he hath reserved in everlasting chains
  under darkness unto the judgment of the great day. 
  
  Revelation 12:8-10 (KJV) And there was war in heaven: Michael and his
  angels fought against the dragon; and the dragon fought and his
  angels, And prevailed not; neither was their place found any more in
  heaven.  And the great dragon was cast out, that old serpent, called
  the Devil, and Satan, which deceiveth the whole world: he was cast out
  into the earth, and his angels were cast out with him.


Regarding the belief that you become an angel when you die, the following scriptures come to mind.


  Romans 8:16-17 The Spirit itself beareth witness with our spirit, that
  we are the children of God: And if children, then heirs; heirs of God,
  and joint-heirs with Christ; if so be that we suffer with him, that we
  may be also glorified together.


If we are joint heirs with Christ, then one day we shall be like him.


  I John  3:2 Beloved, now are we the sons of God, and it doth not yet
  appear what we shall be: but we know that, when he shall appear, we
  shall be like him; for we shall see him as he is.


After his resurrection He was made "better than the angels". If we are to be like Him, then we also will be "better than the angels".


  HEBREWS 1:4-5 Being made so much better than the angels, as he hath by
  inheritance obtained a more excellent name than they. For unto which
  of the angels said he at any time, Thou art my Son, this day have I
  begotten thee? And again, I will be to him a Father, and he shall be
  to me a Son?


The idea that we would become angels (either 'good' or 'fallen') when we die originated from a misinterpretation of Matthew 22:30 and Mark 12:25.


  Matthew 22:30 (KJV) For in the resurrection they neither marry, nor
  are given in marriage, but are as the angels of God in heaven.
  
  MARK 12:25 For when they shall rise from the dead, they neither marry,
  nor are given in marriage; but are as the angels which are in heaven.


The context of these verses is the subject of marriage. Angels do not have wedding ceremonies, and neither will we in heaven. Using these verses as the basis for the belief that we will one day be angels is a bit of a stretch.