Those who repent

Alma 13:46 And since man had fallen, he could not merit anything of himself; but the sufferings and death of Christ atoneth for their sins, through faith and repentance, etc.:

Alma 16:216 This being the intent of this last sacrifice, to bring about the bowels of mercy, which overpowereth justice and bringeth about means unto men that they may have faith unto repentance.

Alma 16:217 And thus mercy can satisfy the demands of justice, and encircles them in the arms of safety, while he that exercises no faith unto repentance, is exposed to the whole law of the demands of justice; therefore, only unto him that has faith unto repentance, is brought about the great and eternal plan of redemption.

Those who have Ignorantly Sinned

Mosiah 1:107 For behold, and also his blood atoneth for the sins of those who have fallen by the transgression of Adam, who have died, not knowing the will of God concerning them, or who have ignorantly sinned.

2 Nephi 6:51-56 Wherefore he hath given a law; and where there is no law given there is no punishment;

52 And where there is no punishment, there is no condemnation;

53And where there is no condemnation, the mercies of the Holy One of Israel have claim upon them, because of the atonement:

54 For they are delivered by the power of him: for the atonement satisfieth the demands of his justice upon all those who have not the law given to them, that they are delivered from that awful monster, death and hell, and the devil, and the lake of fire and brimstone, which is endless torment;

55 And they are restored to that God who gave them breath, which is the Holy One of Israel.

56 But wo unto him that has the law given; yea, that has all the commandments of God, like unto us, and that transgresseth them, and that wasteth the days of his probation; for awful is his state!

Children are covered by the Atonement

Moroni 8:9-25 Behold, I came into the world not to call the righteous, but sinners to repentance; the whole need no physician, but they that are sick; wherefore little children are whole, for they are not capable of committing sin; wherefore the curse of Adam is taken from them in me, that it hath no power over them; and the law of circumcision is done away in me.

10 And after this manner did the Holy Ghost manifest the word of God unto me; wherefore my beloved son, I know that it is solemn mockery before God, that ye should baptize little children.

11 Behold I say unto you, that this thing shall ye teach, repentance and baptism unto those who are accountable and capable of committing sin; yea, teach parents that they must repent and be baptized, and humble themselves as their little children, and they shall all be saved with their little children: and their little children need no repentance, neither baptism. 

13 But little children are alive in Christ, even from the foundation of the world; if not so, God is a partial God, and also a changeable God, and a respecter to persons; for how many little children have died without baptism.

14 Wherefore, if little children could not be saved without baptism, these must have gone to an endless hell.

15 Behold I say unto you, that he that supposeth that little children need baptism, is in the gall of bitterness, and in the bonds of iniquity; for he hath neither faith, hope, nor charity; wherefore, should he be cut off while in the thought, he must go down to hell. 

18 Behold, I speak with boldness, having authority from God; and I fear not what man can do; for perfect love casteth out all fear; and I am filled with charity, which is everlasting love; wherefore all children are alike unto me; wherefore I love little children with a perfect love; and they are all alike, and partakers of salvation. 

20 Little children can not repent; wherefore it is awful wickedness to deny the pure mercies of God unto them, for they are all alive in him because of his mercy.

21 And he that saith that little children need baptism, denieth the mercies of Christ, and setteth at naught the atonement of him and the power of his redemption.

25 For behold that all little children are alive in Christ, and also all they that are without the law.

Those not “Capable” of repentance

Moroni 8:26 For the power of redemption cometh on all they that have no law; wherefore, he that is not condemned, or he that is under no condemnation, can not repent; and unto such baptism availeth nothing.

Moroni 8:28 Behold, my son, this thing ought not to be; for repentance is unto them that are under condemnation, and under the curse of a broken law.

Why is the atonement necessary?

Alma 19:90-97 Therefore, as the soul could never die, and the fall had brought upon all mankind a spiritual death as well as a temporal; that is, they were cut off from the presence of the Lord; therefore it was expedient that mankind should be reclaimed from this spiritual death;
 

91 Therefore as they had been carnal, sensual and devilish, by nature, this probationary state became a state for them to prepare; it became a preparatory state.

92 And now remember, my son, if it were not for the plan of redemption, (laying it aside,) as soon as they were dead, their souls were miserable, being cut off from the presence of the Lord.

93 And now there was no means to reclaim men from this fallen state which man had brought upon himself, because of his own disobedience;

94 Therefore, according to justice, the plan of redemption could not be brought about, only, on conditions of repentance of men in this probationary state; yea, this preparatory state; for except it were for these conditions, mercy could not take effect except it should destroy the work of justice.

97 And now the plan of mercy could not be brought about, except an atonement should be made; therefore God himself atoneth for the sins of the world, to bring about the plan of mercy, to appease the demands of justice, that God might be a perfect, just God, and a merciful God also.

Alma 19:105 But God ceaseth not to be God, and mercy claimeth the penitent, and mercy cometh because of the atonement; and the atonement bringeth to pass the resurrection of the dead: and the resurrection of the dead bringeth back men into the presence of God;