In the Day

Adam was told that he would die in the same day he ate from the tree of the knowledge of good and evil.

And the Lord God commanded the man, saying, Of every tree of the garden thou mayest freely eat: But of the tree of the knowledge of good and evil, thou shalt not eat of it: for in the day that thou eatest thereof thou shalt surely die.
- Genesis 2:16-17 (KJV)

Adam didn't die that same day of course.

And all the days that Adam lived were nine hundred and thirty years: and he died.
- Genesis 5:5 (KJV)

Ellen White however explains it like this:

Adam listened to the words of the tempter, and yielding to his insinuations, fell into sin. Why was not the death penalty at once enforced in his case? - Because a ransom was found. God's only begotten Son volunteered to take the sin of man upon himself, and to make an atonement for the fallen race. There could have been no pardon for sin had this atonement not been made. Had God pardoned Adam's sin without an atonement, sin would have been immortalized, and would have been perpetuated with a boldness that would have been without restraint. Remember how soon after the transgression of Adam the apostasy of his posterity became so marked that God repented that He had made man. They followed the imaginations of their evil hearts, and the strivings of the Spirit were not heeded. They refused to be admonished. They had an abundance of blessings for their own enjoyment, and they soon forgot that they had forfeited immortality.
- Ellen White, The Review and Herald, April 23, 1901

So, in other words, if Christ hadn't intervened that day, Adam would have died.

Either way, Adam was separated from God on that day, and his death was now certain. A bit like a branch that is separated from a tree; it "lives" for a while, and then dies.

Death in this case was a process that started on the day he ate from the tree.