Reason for the Name "Second" Death
The fact that there is something called a "second" death implies that there is something called a "first" death. Such is the case, for the term "first death" is used four times in the scriptures. The "first" death occurs when an individual is cut off temporally from God (that is, separated from Heavenly Father's physical location).
For Adam and Eve, this first death occurred after eating the forbidden fruit: "I, the Lord God, caused that he [Adam] 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" (D&C 29:41). For the rest of us, the first death occurs when we are born into this world, because at that point we are no longer dwelling with God in heaven, but rather are physically separated from him while we dwell on earth.
The use of "first" and "second" may be thought of as chronological, indicating the sequence in which people experience these two types of separation from God. We are first separated from Heavenly Father (the "first death"), when we come to earth as babies. We are later separated from the Holy Ghost (the "second death"), when we become accountable and then sin.
Misconceptions about Why It is Called the "Second" Death
Some assume that the descriptors "first" and "second" simply refer to the fact that there are two types of death, physical and spiritual. By this explanation, "first" might refer to physical death and "second" might refer to spiritual death (or vice versa).
However, such cannot be the case, because the scriptures clearly use both of the terms "first death" and "second death" to refer to aspects of spiritual death. The Lord says mankind "became spiritually dead, which is the first death" (D&C 29:41). Alma says, "Then cometh a death, even a second death, which is a spiritual death" (Alma 12:16). Samuel the Lamanite also clearly uses both terms to refer to spiritual death: "[The atonement] redeemeth all mankind from the first death—that spiritual death. ... Whosoever repenteth not is hewn down and cast into the fire; and there cometh upon them again a spiritual death, yea, a second death" (Hel. 14:16, 18).
Thus, neither "first death" nor "second death" refers to physically dying; both terms refer to aspects of spiritual death, or separation from God.
Effects in This Life
In a technical sense, the second death means to be cut off from the Holy Ghost as a result of sin. By that definition, any of us mortals who have reached the age of accountability and sinned have experienced the second death---at least to a small degree. We lose the influence of the Holy Ghost and of the light of Christ in increments, to the degree that we ignore the commandments and rebel against God's will. While members often use "second death" to mean being permanently separated from God after the Judgment, to be accurate we must acknowledge that we can begin tasting aspects of the second death in this life.
This is comparable to the way we sometimes talk about physical death. Even while someone is alive, before they have permanently physically died, we say that death is "knocking at their door." When someone undergoes great pain or illness, that is in one sense the effects of physical death upon that person. Thus, physical death is often thought of as a one-time, permanent event, but it can also be thought of as a continuing state or condition before the event. Likewise, the second death is often thought of as a one-time, permanent event, but it is sometimes used in the scriptures as a continuing state or condition before the Judgment.
Effect in the Next Life
While the second death technically refers to losing the Holy Ghost as a result of sin, even in this life, usually the scriptures focus on what happens after the Judgment to those who don't repent. They speak of not merely tasting a sampling of the second death in this mortal life, but of being overwhelmed by it in the next. For example, they talk of being "brought down unto," "hurt of," or "cast into" the second death (Hel. 14:19; Rev. 2:11; Jacob 3:11). This means that while the second death merely affects us in this life, in the next life it will "have power over" the unrepentant (Rev. 20:6, 14; D&C 76:37). In this regard, since everyone else will dwell in God's presence to a degree, the sons of perdition are "the only ones on whom the second death shall have any power" (D&C 76:37).
The scriptures describe the second death as being "cut off again as to things pertaining to righteousness" (Hel. 14:18). Because of the atonement, all mankind unconditionally overcomes the first spiritual death and returns to God's presence: the Savior "redeemeth all mankind from the first death—that spiritual death, ... yea, even all mankind, and bringeth them back into the presence of the Lord" (Hel. 14:16--17); "All men come unto God; wherefore, they stand in the presence of him, to be judged" (2 Ne. 2:10). Those who have not met the conditions of repentance will then be separated from God a second time---the second death.
One distinguishing feature of the second death is mentioned by Alma: it is "an everlasting death as to things pertaining unto righteousness" (Alma 12:32). In other words, it is permanent. When a person leaves God's presence after the final Judgment, "never at any time have I declared from mine own mouth that they should return, for where I am they cannot come, for they have no power" (D&C 29:29). Bold text