What are the various eternal destinies or after lives that are possible in LDS Mormon Doctrine?

There are four major different eternal destinations after this life according to Mormon doctrine:

• The celestial (heavenly) kingdom or glory. Those who have faith in Jesus Christ and obey his gospel (see Doctrine and Covenants 76:50-70).

• The terrestrial (earthly) kingdom or glory. Those who are honorable people but have not proven faithful enough. For more details see Doctrine and Covenants 76:71-80.

• The telestial (distant) kingdom or glory. Very roughly, those are the bad people. Yet the glory of the telestial kingdom "surpasses all understanding" (see Doctrine and Covenants 76:81-90,98-106).

• Outer darkness. Those who commit the unpardonable sin of denying the Holy Ghost after having received a full knowledge and a perfect witness of the truthfulness of the gospel including Satan and his fallen angels (see Doctrine and Covenants 76:25-39). Only very few people living on the earth will end up there.

I only wrote roughly who goes where, but in the given references it explains more detailed the requirements and the state of these glories. The most interesting place is the celestial kingdom. Accoding to LDS doctrine the gospel of Jesus Christ requires one to

• have faith in Jesus Christ and His atonement,
• repent of their sins,
• be baptized by immersion,
• receive the gift of the Holy Ghost by the laying on of hands and
• keeping the covenants they have entered into until the end

in order to enter the celestial kingdom. There are many who did not have the chance to do these things. Here the following scripture applies (see Doctrine and Covenants 137:7-10):

7 [...] All who have died without a knowledge of this gospel, who would have received it [...], shall be heirs of the celestial kingdom of God;

[...]

10 [...] all children who die before they arrive at the years of accountability are saved in the celestial kingdom of heaven.

Therefore, we do not baptize little children until they can make a conscious decision.

Furthermore the celestial glory itself has three degrees (see Doctrine and Covenants 131:1-2):

1 In the celestial glory there are three heavens or degrees;

2 And in order to obtain the highest, a man must enter into this order of the priesthood [meaning the new and everlasting covenant of marriage];

In practice this means that the highest degree in the celestial kingdom can only be received, if a man and a women are sealed (married) in an LDS temple. The same principle as in Doctrine and Covenants 137:7-10 applies here.

LDS doctrine on the subject is found in section 76 of the Doctrine and Covenants. Referring to the circumstances in which he received this revelation, Joseph Smith stated:

From sundry revelations which had been received, it was apparent that many important points touching the salvation of man had been taken from the Bible, or lost before it was compiled. It appeared self-evident from what truths were left, that if God rewarded every one according to the deeds done in the body the term ‘Heaven,’ as intended for the Saints’ eternal home, must include more kingdoms than one.

Upon seeking clarification from the Lord, this revelation was received, regarding the nature of the eternal destiny of mankind. It first speaks of the "sons of perdition," people whose choices have placed them outside of the Atonement.

31 Thus saith the Lord concerning all those who know my power, and have been made partakers thereof, and suffered themselves through the power of the devil to be overcome, and to deny the truth and defy my power—

32 They are they who are the sons of perdition, of whom I say that it had been better for them never to have been born;

33 For they are vessels of wrath, doomed to suffer the wrath of God, with the devil and his angels in eternity;

34 Concerning whom I have said there is no forgiveness in this world nor in the world to come—

35 Having denied the Holy Spirit after having received it, and having denied the Only Begotten Son of the Father, having crucified him unto themselves and put him to an open shame.

36 These are they who shall go away into the lake of fire and brimstone, with the devil and his angels—

37 And the only ones on whom the second death shall have any power;

38 Yea, verily, the only ones who shall not be redeemed in the due time of the Lord, after the sufferings of his wrath.

So the sons of perdition are people who "know my power and have been made partakers thereof" and then "deny the truth and defy my power," and they will be consigned to hell for eternity, with Satan and his fallen angels.

After this, it moves on to a more pleasant topic, the fate of those who have not committed unpardonable sins, and defines three "degrees of glory," as they're commonly known in LDS theology:

50 And again we bear record—for we saw and heard, and this is the testimony of the gospel of Christ concerning them who shall come forth in the resurrection of the just—

51 They are they who received the testimony of Jesus, and believed on his name and were baptized after the manner of his burial, being buried in the water in his name, and this according to the commandment which he has given—

52 That by keeping the commandments they might be washed and cleansed from all their sins, and receive the Holy Spirit by the laying on of the hands of him who is ordained and sealed unto this power;

53 And who overcome by faith, and are sealed by the Holy Spirit of promise, which the Father sheds forth upon all those who are just and true.

...

58 Wherefore, as it is written, they are gods, even the sons of God—

59 Wherefore, all things are theirs, whether life or death, or things present, or things to come, all are theirs and they are Christ’s, and Christ is God’s.

60 And they shall overcome all things.

...

70 These are they whose bodies are celestial, whose glory is that of the sun, even the glory of God, the highest of all, whose glory the sun of the firmament is written of as being typical.

The reward of the most faithful is commonly known as "the Celestial kingdom," or also "exaltation," and is the idea that people unfamiliar with LDS theology are thinking of when they hear that Mormons believe they can become Gods in the afterlife.

After this, the revelation speaks of the the eternal rewards for people who have not lived a Celestial-grade life:

71 And again, we saw the terrestrial world, and behold and lo, these are they who are of the terrestrial, whose glory differs from that of the church of the Firstborn who have received the fulness of the Father, even as that of the moon differs from the sun in the firmament.

72 Behold, these are they who died without law;

73 And also they who are the spirits of men kept in prison, whom the Son visited, and preached the gospel unto them, that they might be judged according to men in the flesh;

74 Who received not the testimony of Jesus in the flesh, but afterwards received it.

75 These are they who are honorable men of the earth, who were blinded by the craftiness of men.

76 These are they who receive of his glory, but not of his fulness.

77 These are they who receive of the presence of the Son, but not of the fulness of the Father.

78 Wherefore, they are bodies terrestrial, and not bodies celestial, and differ in glory as the moon differs from the sun.

79 These are they who are not valiant in the testimony of Jesus; wherefore, they obtain not the crown over the kingdom of our God.

Those who inherit "the Terrestrial kingdom" can be thought of as people who were "good, but not great," and thus their reward was not as great as those who were more valiant in their faith.

And finally, it speaks of "the Telestial kingdom," the lowest of the three degrees of glory:

81 And again, we saw the glory of the telestial, which glory is that of the lesser, even as the glory of the stars differs from that of the glory of the moon in the firmament.

82 These are they who received not the gospel of Christ, neither the testimony of Jesus.

83 These are they who deny not the Holy Spirit.

84 These are they who are thrust down to hell.

85 These are they who shall not be redeemed from the devil until the last resurrection, until the Lord, even Christ the Lamb, shall have finished his work.

86 These are they who receive not of his fulness in the eternal world, but of the Holy Spirit through the ministration of the terrestrial;

...

99 For these are they who are of Paul, and of Apollos, and of Cephas.

100 These are they who say they are some of one and some of another—some of Christ and some of John, and some of Moses, and some of Elias, and some of Esaias, and some of Isaiah, and some of Enoch;

101 But received not the gospel, neither the testimony of Jesus, neither the prophets, neither the everlasting covenant.

102 Last of all, these all are they who will not be gathered with the saints, to be caught up unto the church of the Firstborn, and received into the cloud.

103 These are they who are liars, and sorcerers, and adulterers, and who*emongers, and whosoever loves and makes a lie.

104 These are they who suffer the wrath of God on earth.

105 These are they who suffer the vengeance of eternal fire.

106 These are they who are cast down to hell and suffer the wrath of Almighty God, until the fulness of times, when Christ shall have subdued all enemies under his feet, and shall have perfected his work;

...

111 For they shall be judged according to their works, and every man shall receive according to his own works, his own dominion, in the mansions which are prepared;

112 And they shall be servants of the Most High; but where God and Christ dwell they cannot come, worlds without end.

People who have not lived righteous lives, and yet have not committed the unforgivable sin, will eventually be forgiven after they have suffered for their sins, and receive a small reward in heaven according to the quality of the lives they chose to live.

Section 76 is one of the longest chapters in the Doctrine and Covenants, and I've cut down the quotes a fair bit for brevity, but this covers the core of the LDS doctrine on the afterlife and the various different destinies available to mankind.