The death promised and that Adam experienced the moment he chose to sin was not a ceasing to exist or a physical stopping to breathe, it was a separation: a rending apart of two things that were once closely joined. On the day that Adam ate the fruit, he died1. He was separated from God. The fellowship between them was broken.
The reverse of this death happens when we are 'born again'. The fellowship is restored, and we are born into new life -- the power of the cross is that it reverses death!
The alternative, for those who are not born again into a new life, is that they will experience a 'second death'. This is partly a contrast between physical and spiritual death. It happens sequentially after physical death (making it second) but is also the point at which spiritual separation becomes permanent (having died once with Adam, they will be judged to permanent death). The reverse is true for believers: we will be raised into life even though we die.
This is an important doctrine to understand in light of many popular culture understandings of death as some kind of annihilation or ceasing to exist. Never in Scripture do we find an indications that any humans well cease to exist, but we do find lots of information about the change in our relationship before God.
This understanding of death as a separation or rending apart of two things once joined is also the sense of 'death' that Romans and 1st Peter use when they speak of believers being dead to sin. We are no longer bound to it as a slave is to his master. Believers relationship with sin has been severed and we have been cleaved to a new master.
When we physically die, our spirits are separated from the bodies that we have been one with2. Our bodies don't cease to exist nor do our spirits, but they part ways3. Our bodies then decay and return to dust, but even the material in this world doesn't cease to exist. Our spirits meanwhile continue, either in eternal life (unity with Christ) or eternal death (separation from Him).
Genesis 2:17 (ESV)
but of the tree of the knowledge of good and evil you shall not eat, for in the day that you eat1 of it you shall surely die.”
Genesis 35:18 (ESV)
And as her soul was departing (for she was dying), she called his name Ben-oni; but his father called him Benjamin.
James 2:26 (ESV)
For as the body apart from the spirit is dead, so also faith apart from works is dead.
Ecclesiastes 12:7 (ESV)
and the dust returns to the earth as it was, and the spirit returns to God who gave it.