A fine question to answer on Christmas Eve, I suppose. Revisiting these doctrines is a good way to remind myself why Christ was born: to save us and all Creation.

This is a detailed, but only introductory, answer for those interested in a thorough study of Mormon doctrine. If you're just looking for a basic answer to "How are we saved?" I recommend Confutus' answer or some articles on Mormon.org.

In LDS theology, being saved is an ambiguous term. In the general sense, we are all saved through Christ's Atonement (His suffering, death, and resurrection). Everyone, even the wicked, will be resurrected so they can be brought back into the presence of God to be judged. Amulek explains this in the Book of Mormon, in Alma 11:40-41 (read through 45):

40 And he shall come into the world to redeem his people; and he shall take upon him the transgressions of those who believe on his name; and these are they that shall have eternal life, and salvation cometh to none else.

41 Therefore the wicked remain as though there had been no redemption made, except it be the loosing of the bands of death; for behold, the day cometh that all shall rise from the dead and stand before God, and be judged according to their works.

Here, there is clearly a distinction between the group that is resurrected (everyone), and the group that obtains eternal life (those who believe on His name).

The resurrection - being saved from the bands of death - is made possible because of Christ's resurrection and it is by His power that we are redeemed from the grave, as explained in D&C 88:14-17:

14 Now, verily I say unto you, that through the redemption which is made for you is brought to pass the resurrection from the dead.

15 And the spirit and the body are the soul of man.

16 And the resurrection from the dead is the redemption of the soul.

17 And the redemption of the soul is through him that quickeneth all things, in whose bosom it is decreed that the poor and the meek of the earth shall inherit it.

It goes on, referring to the new Earth in Revelation 21 you wrote about in your question:

18 Therefore, it must needs be sanctified from all unrighteousness, that it may be prepared for the celestial glory;

19 For after it hath filled the measure of its creation, it shall be crowned with glory, even with the presence of God the Father;

20 That bodies who are of the celestial kingdom may possess it forever and ever; for, for this intent was it made and created, and for this intent are they sanctified.

This is where we get to the specific meaning of the term "saved" like in your question: How do we obtain the celestial glory with which God lives?

Doctrine and Covenants 76 records a vision of who obtain the celestial kingdom:

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.

(I've emphasized the saving ordinances referred to below.)

Most of these characteristics (keeping commandments, receive the Holy Spirit, etc.) should be familiar with Christianity at large. Granted, "The Holy Spirit of promise" is probably an unfamiliar term, but its explanation would deserve its own question.

In any case, it's a tall order to qualify for this salvation, but it's nothing less than what would be expected of a God who wants us to reach our potential.

Requirements for salvation are not dependent on race or gender. They are universal no matter who, what, where, or when you are.

If you're a "non-Mormon" you are neither exempt nor disqualified for the celestial kingdom. It is not a matter of being a "good Mormon" or an active member of "the right church" (both are common phrases, but not with Latter-day Saints; they come off kind of snobby) to be saved, in that sense. It is more about receiving the necessary saving ordinances by proper authority by which we make sacred covenants.

So in practice, how is this achieved? By consistently being who/what we are trying to be come, by following the example of Jesus Christ and being sanctified by His Spirit. This is what Mormons focus on in daily life. And for those who haven't yet been baptized, there are the LDS missionaries to help with that.

The Book of Mormon concludes with words from Moroni that emphasize this gradual sanctification:

32 Yea, come unto Christ, and be perfected in him, and deny yourselves of all ungodliness; and if ye shall deny yourselves of all ungodliness, and love God with all your might, mind and strength, then is his grace sufficient for you, that by his grace ye may be perfect in Christ; and if by the grace of God ye are perfect in Christ, ye can in nowise deny the power of God.

33 And again, if ye by the grace of God are perfect in Christ, and deny not his power, then are ye sanctified in Christ by the grace of God, through the shedding of the blood of Christ, which is in the covenant of the Father unto the remission of your sins, that ye become holy, without spot.

The simplest answer to this is given in the Articles of Faith:

3 We believe that through the Atonement of Christ, all mankind may be saved, by obedience to the laws and ordinances of the Gospel.

4 We believe that the first principles and ordinances of the Gospel are: first, Faith in the Lord Jesus Christ; second, Repentance; third, Baptism by immersion for the remission of sins; fourth, Laying on of hands for the gift of the Holy Ghost."

An additional principle made clear in scripture but not cited in this summary is that it is necessary to "endure to the end" in exercising faith, repenting, and obedience to the commandments of God. The LDS consider salvation to be a process more than an event, a process which is usually not complete until one has died. (This is why they typically have difficulty giving a simple "yes" or "no" answer to the question "Have you been saved?")

Race and gender do not matter. The degree of accountability or responsibility a person has does. In addition,

Doctrine and Covenants 137:7-10

"All who have died without a knowledge of this gospel who would have received it if they had been permitted to tarry, shall be heirs of the celestial kingdom of God. Also, all those who shall die henceforth, who would have received it with all their hearts, shall be heirs of that kingdom. For I, the Lord, will judge all man according to their works, according to the desire of their hearts. And I also beheld that all children who die before they arrive at the years of accountability are saved in the celestial kingdom of heaven".

Most references in the scriptures to salvation refer to salvation in the celestial kingdom. Two other degrees, and the requirements for each of these degrees, are described in Doctrine and Covenants section 76. There is also a state of no salvation described there.

Faith in Jesus Christ is understood to include acceptance of his latter-day prophets and apostles (Joseph Smith and successors) and the additional Testament he has given (the Book of Mormon) as well as the Old Testament and the New Testament. Anyone may repent and all are commanded to, but repentance must be genuine and involve a permanent change of conduct, not lip service only. The ordinances of baptism and the laying on of hands for the gift of the Holy Ghost must be performed by those who have been ordained to the same priesthood given to Joseph Smith and according to the order of the Church.