What are the different types of death in the Bible?

Physical death occurs when the spiritual body (soul and spirit) separates from the body of flesh and blood. 

Genesis 35:18 describes this as such: 

"And it came to pass, as her soul was in departing, (for she died) that she called his name Ben-oni: but his father called him Benjamin." 

Physical death does not result in becoming nonexistent; it is a separation of the human spirit and soul from the physical body. Again, death is separation.

The other is spiritual death, when the spirit of man (as an individual) is unable to commune with the Spirit of God. Here again is a type of separation: man is spiritually separated from his Creator. There is no unity or fellowship between the two. You may read 1 Corinthians 2:9-16 to learn how sin has made the spirit of man incompatible with the Spirit of God.

There is also positional death. As people who have trusted Jesus Christ's death, burial, and resurrection as sufficient payment for our sins (1 Corinthians 15:3-4); we have been crucified with Him and been raised again with Him. 

Romans chapter 6 speaks about positional death: 

1 What shall we say then? Shall we continue in sin, that grace may abound? 2 God forbid. How shall we, that are dead to sin, live any longer therein? 3 Know ye not, that so many of us as were baptized into Jesus Christ were baptized into his death? 4 Therefore we are buried with him by baptism into death: that like as Christ was raised up from the dead by the glory of the Father, even so we also should walk in newness of life. 5 For if we have been planted together in the likeness of his death, we shall be also in the likeness of his resurrection: 6 Knowing this, that our old man is crucified with him, that the body of sin might be destroyed, that henceforth we should not serve sin. 7 For he that is dead is freed from sin.

We are "dead to sin" in that we are no longer "in Adam." God has reckoned us separated from Adam's damnation; we are now a "new creature in Christ" (2 Corinthians 5:17). Through personal faith in Christ's finished cross work, our position "in Adam" has been exchanged for our position "in Christ." Now, we have the ability to be vessels of God's power and exhibitions of His grace!

There is one final "death" (separation) in the Bible. We have covered functional death, physical death, spiritual death, and positional death. The last "death" is "the second death," and it is connected to spiritual death. While a lost or unsaved soul on Earth is spiritually dead—separated from God's life—that can be remedied by personal faith in the Gospel of Grace (1 Corinthians 15:3-4). If that spiritual death is not replaced with eternal life before physical death arrives, that spiritual death will be replaced with the second death. As the spiritual body separates from the physical body, now the spiritual body separates from God's Spirit forever. All hope of ever gaining a right standing before God is now permanently gone. The second death occurs when the lost souls in Hell are judged, and then tossed into the Lake of Fire.

Revelation chapter 20 mentions "the second death:" 

11 And I saw a great white throne, and him that sat on it, from whose face the earth and the heaven fled away; and there was found no place for them. 12 And I saw the dead, small and great, stand before God; and the books were opened: and another book was opened, which is the book of life: and the dead were judged out of those things which were written in the books, according to their works. 13 And the sea gave up the dead which were in it; and death and hell delivered up the dead which were in them: and they were judged every man according to their works. 14 And death and hell were cast into the lake of fire. This is the second death. 15 And whosoever was not found written in the book of life was cast into the lake of fire.

Also, Revelation 21:8: 

But the fearful, and unbelieving, and the abominable, and murderers, and whoremongers, and sorcerers, and idolaters, and all liars, shall have their part in the lake which burneth with fire and brimstone: which is the second death.

An extract from an article by Shawn Brasseaux called: "Exactly what is eternal life"