God said to him that he would "surely die" if he ate of the forbidden fruit. But Adam lived for hundreds of years after that. So, what died in Adam? Remember the first thing mentioned right after he'd sinned against God was that he tried to hide from God because he was afraid. Before that God had asked him to name all the animals. How was it that he was all of a sudden afraid of God?
Well, God made man in his own image (God's image is Holy Spirit); he also made man from the earth (body); he also breathed soul life into man and he became a living soul (breath life). What happened was that Adam lost the Holy Spirit part of him, that is what "died". Holy Spirit is required for man to communicate with God. When Adam lost that he became afraid of God. Can you imagine the fear he had?