The best way to understand this would be to study the story found in the book of Genesis in the Holy Bible. Let us begin with a brief outline of the story. We are told that God warned Adam that if he eats from the tree of the knowledge of good and evil, he will surely die. This was communicated also to Eve. The serpent, however, being deceptive, promised Eve that they will not die from eating it. So she ate the fruit, and gave also to her husband. As a result of eating the fruit, they realized for the first time that they were naked, so they covered themselves with fig leaves. When they heard the sound of God walking in the garden in the cool of the day, they hid from Him among the trees. He called out to them asking where they were and asked whether they had eaten from the forbidden tree. Adam blamed Eve, and Eve blamed the serpent. God then cursed all three of them. The serpent shall henceforth crawl on its belly and eat dust. The woman shall suffer in childbearing and remain under the domination of her husband. The man shall have to sweat for his living until he returns to dust. God then said: "The man has now become like one of us, knowing good and evil. He must not be allowed to reach out his hand and take also from the tree of life and eat, and live forever." So God banished him from the garden and placed a flaming sword to prevent access to the tree of life (Condensed from Genesis 2:17 - 3:22).
Many people misunderstand this story to mean that physical death is a consequence for sin. But the biblical scholars who are aware of all its implications explain otherwise. The Interpreter's One Volume Commentary on the Bible explains that death here means separation from God, the giver of "life" (p.6).
The reason for this other explanation is the obvious fact that in the story Adam did not die. He lived on for 930 years (see Genesis 5:5). And when God spoke of his eventual death, He mentioned it not as a consequence of sin, but as a natural outcome of the fact that Adam was created from dust - and to dust he must return (see Genesis 3:19).
When a few verses later it is stated that God deprived Adam in access to the tree of life many people again conclude that because of sin Adam was deprived of everlasting life. But this too is not what the book of Genesis says. The story goes that once Adam gained knowledge of good and evil, God was concerned that he may now gain eternal life also (see Genesis 3:22). The Abingdon Bible Commentary explains that Adam was deprived of eternal life because he had already acquired some power, namely knowledge, and God was concerned lest Adam should acquire more, namely eternal life, and become a threat to God (see the Abingdon Bible Commentary, p.223).
It is clear from the story that even if Adam was promised death he was given a lesser penalty, and death must be taken as the maximum possible penalty - that, obviously, was not given. Elsewhere in the book of Genesis, when God killed certain men for their wickedness, it meant instant death (see Genesis 38: 7, 10). But the death that comes as the natural end to physical life is not a penalty for sin. It is better to understand that God in His mercy gave a lesser penalty to Adam than to suppose that God in His anger gave more penalties in addition to what He stated initially. This is why Reverend W. K. Lowther Clarke explained in his Bible commentary that God relented and gave Adam a lesser penalty (see the Concise Bible Commentary 1952, p.343). For the Qur'anic perspective on the Genesis story, see question 17.