The second death, or spiritual death, is defined as being cut off from the presence of God. This means that you cannot see the face of God, you cannot withstand the presence of God, you will not enjoy the visitation of the Son, and you are not worthy of the smallest portion of the spirit of the Holy Ghost. It is a total and complete separation from God. As described in the scripture, this state happens to those who suffer in hell. It also describes the state of the Sons of Perdition who are cast out after the last soul is resurrected. Therefore, the second death has power upon all those who suffer in hell until the resurrection of the wicked. At that point the second death applies only to the Sons of Perdition, These are they who shall go away into the lake of fire and brimstone, with the devil and his angels—And the only ones on whom the second death shall have any power (DC 76:36-7).
One might be inclined to think that the “first death” must be the death of the mortal body. This is not the case. The Lord explains that the first death is also spiritual and occurs when we leave his presence and come to earth. This is symbolized by the fall of Adam, I, the Lord God, caused that he should be cast out from the Garden of Eden, from my presence, because of his transgression, wherein he became spiritually dead, which is the first death, even that same death which is the last death (or second death), which is spiritual, which shall be pronounced upon the wicked when I shall say: Depart, ye cursed (DC 29:41). Our mortal death is a mere byproduct of the first spiritual death. Hence, Samuel states we are considered as dead, both as to things temporal and to things spiritual, by virtue of the fact that we have been cut off from the Lord’s presence.
Joseph F. Smith
“But I want to speak a word or two in relation to another death, which is a more terrible death than that of the body. When Adam, our first parent, partook of the forbidden fruit, transgressed the law of God, and became subject unto Satan, he was banished from the presence of God and was thrust out into outer spiritual darkness. This was the first death. Yet living, he was dead--dead to God, dead to light and truth, dead spiritually; cast out from the presence of God; communication between the Father and the son cut off. He was as absolutely thrust out from the presence of God as was Satan and the hosts that followed him. That was spiritual death. But the Lord said that He would not suffer Adam nor his posterity to come to the temporal death until they should have the means by which they might be redeemed from the first death, which is spiritual. Therefore angels were sent unto Adam, who taught him the Gospel and revealed to him the principle by which he could be redeemed from the first death, and be brought back from banishment and outer darkness into the marvelous light of the Gospel. He was taught faith, repentance and baptism for the remission of sins, in the name of Jesus Christ, who should come in the meridian of time and take away the sin of the world, and was thus given a chance to be redeemed from the spiritual death before he should die the temporal death.” (Conference Report, Oct. 1899, p. 72)
Russell M. Nelson
"Therefore, spiritual (or the second) death becomes a matter of great importance to us while we sojourn here in mortality. Even the Savior cannot save individuals in their sins. He will redeem them from their sins, but then only through their repentance. Only rarely may we be responsible for physical death, but we are solely responsible for spiritual death.
“Whether spiritual death be numbered as the second or the first death is incidental. The real objective is to avoid it.” (The Gateway We Call Death, p. 14)