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)

But Adam was separated from God on that day, and his death was now certain. (Like a separated branch from a tree)

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

The verse also says "thou shalt surely die", and not "thou shall surely die instantly".

I hope that makes sense.