What do mormons believe about eschatology?

LDS eschatology is laid out in Doctrine and Covenants section 76. It is a rather unique doctrine among Christianity, in that it explicitly rejects the common idea of "one heaven and one hell." In the header, it describes the context in which Joseph Smith received this revelation:

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.

The revelation describes that there are three kingdoms of Heaven, of varying degrees of glory, that the Sons of Perdition--"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"--will be cast into outer darkness and eternal torment alongside the devil and his angels, but all others, including ordinary wicked people, will receive some degree of heaven after having been punished for their sins in the interim period between death and resurrection.

It contains an interesting statement regarding the heirs of the lowest kingdom, comprised of the "ordinary bad people" of the Earth, if you will:

84 These are they who are thrust down to hell.

This seems to imply that, though they do receive some eternal reward, they are still considered "condemned" in the sense that there is so much more glory that they could have attained, but they failed to recognize that potential through their choices.

It is based on obedience and what is in the heart. One scripture that brings this out is:

"For we labor diligently to write, to persuade our children, and also our brethren, to believe in Christ, and to be reconciled to God; for we know that it is by grace that we are saved, after all we can do." (2 Nephi 25:23)

It has been stressed that it is not "we are saved because of what we do" but rather "after all we can do" ("The Gift of Grace", President Dieter F. Uchtdorf, 2015 April Conference, Sunday morning session). What can "we do"? We can love God with all our heart, mind, and soul (Matthew 22:26-40). We can then show that love through obedience to the commandments (John 14:15). We can repent of our sins ("After All We Can Do", Elder Claudio D. Zivic of the Seventy, 2007 October General Conference). What it really boils down is to strive to measure up to the stature of Jesus Christ (Ephesians 4:11-13) by overcoming the "natural man":

"For the natural man is an enemy to God, and has been from the fall of Adam, and will be, forever and ever, unless he yields to the enticings of the Holy Spirit, and putteth off the natural man and becometh a saint through the atonement of Christ the Lord, and becometh as a child, submissive, meek, humble, patient, full of love, willing to submit to all things which the Lord seeth fit to inflict upon him, even as a child doth submit to his father." (Mosiah 3:19)

The LDS beliefs align with Universalism such as is expressed in these Articles of Faith:

11 We claim the privilege of worshiping Almighty God according to the dictates of our own conscience, and allow all men the same privilege, let them worship how, where, or what they may.

12 We believe in being subject to kings, presidents, rulers, and magistrates, in obeying, honoring, and sustaining the law.

13 We believe in being honest, true, chaste, benevolent, virtuous, and in doing good to all men; indeed, we may say that we follow the admonition of Paul—We believe all things, we hope all things, we have endured many things, and hope to be able to endure all things. If there is anything virtuous, lovely, or of good report or praiseworthy, we seek after these things.

One point that may differ slightly from Universalism may be that the church itself remains politically neutral and will not allow the buildings to be used for political purposes. However members are strongly encouraged to be active in the political process.

There is no doctrine on Annihilationism that I am aware of though some opinions have been tossed about regarding it. As far as the final judgment goes there will be those that are sent to the Lake of Fire:

"30 And we saw a vision of the sufferings of those with whom he made war and overcame, for thus came the voice of the Lord unto us:

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." (D&C 76:30-38)

The Eternal-Conscious-Torment belief is held by some based on the following verse while others hold to the previous verse in D&C 76.

"Therefore if that man repenteth not, and remaineth and dieth an enemy to God, the demands of divine justice do awaken his immortal soul to a lively sense of his own guilt, which doth cause him to shrink from the presence of the Lord, and doth fill his breast with guilt, and pain, and anguish, which is like an unquenchable fire, whose flame ascendeth up forever and ever." (Mosiah 2:38)

The Bible Dictionary defines "hell" this way:

An English translation of the Hebrew word Sheol, hell signifies an abode of departed spirits and corresponds to the Greek Hades. In common speech it generally denotes the place of torment for the wicked, although it has been often held, both in the Jewish and the Christian churches, that Hades (meaning broadly the place of all departed spirits) consists of two parts, paradise and Gehenna, one the abode of the righteous and the other of the disobedient. Gehenna, or Gehenna of fire, is the Greek equivalent of the “valley of Hinnom,” a deep glen of Jerusalem where the idolatrous Jews offered their children to Moloch (2 Chr. 28:3; 33:6; Jer. 7:31; 19:2–6). It was afterwards used as a place for burning the refuse of the city (2 Kgs. 23:10) and in that way became symbolic of the place of torment (Matt. 5:22, 29–30; 10:28; 18:9; 23:15, 33; Mark 9:43, 45, 47; Luke 12:5; James 3:6). Expressions about “hell-fire” are probably due to the impression produced on men’s minds by the sight of this ceaseless burning and are figurative of the torment of those who willfully disobey God.

In latter-day revelation hell is spoken of in at least two senses. One is the temporary abode in the spirit world of those who were disobedient in this mortal life. It is between death and the Resurrection, and persons who receive the telestial glory will abide there until the last resurrection (D&C 76:84–85, 106), at which time they will go to the telestial glory. In this sense the Book of Mormon speaks of spiritual death as hell (2 Ne. 9:10–12). Hell, as thus defined, will have an end, when all the captive spirits have paid the price of their sins and enter into a degree of glory after their resurrection. Statements about an everlasting hell (Hel. 6:28; Moro. 8:13) must be interpreted in their proper context in the light of D&C 19:4–12, which defines eternal and endless punishment.

On the other hand, the devil and his angels, including the sons of perdition, are assigned to a place spoken of as a lake of fire—a figure of eternal anguish. This condition is sometimes called hell in the scriptures (2 Pet. 2:4; D&C 29:38; 88:113). This kind of hell, which is after the Resurrection and Judgment, is exclusively for the devil and his angels and is not the same as that consisting only of the period between death and resurrection. The one group are redeemed from hell and inherit some degree of glory. The other receive no glory. They continue in spiritual darkness. For them the conditions of hell remain. [https://www.lds.org/scriptures/bd/hell?lang=eng&letter=H]