Death of the body was not God's original plan. It is the result of sin. When we die, we will also face judgment before God. The Bible speaks of death as an enemy. All of these have been brought on mankind because of sin and a fallen race.
Death is an evil that produces torment, fear, hatred, suffering, agony, pain, grief and heartache.
The death, burial and resurrection of the Lord Jesus Christ, God's Son, has now made a way for us to also conquer death. This is the death Jesus is referring to in the above verse, and it is ultimately the death that He came to deliver us from. This death is eternal separation from God. It extends beyond our mortal bodies. It does not mean ceasing to exist, rather it means that we will exist forever in our sinful state.