According to the LDS Church, what is the purpose or result of proxy baptisms for those who have already died?

This is best answered from the Church's statement on this topic:

"Because all who have lived on the earth have not had the opportunity to be baptized by proper authority during life on earth, baptisms may be performed by proxy, meaning a living person may be baptized in behalf of a deceased person.

"Some have misunderstood that when baptisms for the dead are performed the names of deceased persons are being added to the membership records of the Church. This is not the case.

I hope this clears up any questions you have. You can follow the link to see more.

Great question, Narnian; great answer, Mason. Very complete.

I think, Narnian, you had a great question at the end that I'd like to respond to:

If someone is left out by an oversight or who had no children, does that person simply get deprived of something because they were overlooked?

Short answer: no. We know God is not capricious, and he is no respecter of persons, which means that every person born on the earth will have an equal opportunity to receive all of the saving ordinances.

In the final judgment, every child of God will have had a chance to receive all the necessary ordinances, whether in life or after death by proxy. That's the great beauty of the gospel: that the rules are absolute (justice), and that a way will be provided for absolutely everybody to follow the rules (mercy).

Please allow me to link to a discussion on this subject: http://www.jefflindsay.com/LDSFAQ/FQ_BaptDead.shtml#why

Narnian, you also asked this:

I also wonder if proxy baptisms are made for children who die prior to birth, either by abortion or miscarriage.

Let me defer to the Book of Mormon prophet Mormon's discussion of infant baptism:

8 Listen to the words of Christ, your Redeemer, your Lord and your God. 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.

9 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.

Again, this question of baptism of little children (and those that die before birth), also deserves its own question on this site. But for the sake of answering this question, may I recommend reading all of Moroni 8, from which the above quote was taken. And then, may I suggest that if little children are not capable of sin and therefore need no repentance and no baptism, then surely children who die prior to birth also need neither repentance nor baptism.

There is surely more to be discussed here, but hopefully this is a good start.

To understand the doctrine of baptism for the dead, it's necessary to first understand the doctrine of baptism, and the crucial role it plays in LDS theology. Baptism is held to be essential for the salvation of everyone with the maturity to be capable of committing sin. The fourth Article of Faith states:

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.

This is consistent with Acts chapter 2, in which, after hearing the Apostles' miraculous teachings:

37 Now when they heard this, they were pricked in their heart, and said unto Peter and to the rest of the apostles, Men and brethren, what shall we do?

38 Then Peter said unto them, Repent, and be baptized every one of you in the name of Jesus Christ for the remission of sins, and ye shall receive the gift of the Holy Ghost.

The same principle is found in Acts 19:

1 And it came to pass, that, while Apollos was at Corinth, Paul having passed through the upper coasts came to Ephesus: and finding certain disciples,

2 He said unto them, Have ye received the Holy Ghost since ye believed? And they said unto him, We have not so much as heard whether there be any Holy Ghost.

3 And he said unto them, Unto what then were ye baptized? And they said, Unto John's baptism.

4 Then said Paul, John verily baptized with the baptism of repentance, saying unto the people, that they should believe on him which should come after him, that is, on Christ Jesus.

5 When they heard this, they were baptized in the name of the Lord Jesus.

6 And when Paul had laid his hands upon them, the Holy Ghost came on them; and they spake with tongues, and prophesied.

7 And all the men were about twelve.

This poses a problem, though: if God wishes to have all men be saved and come to a knowledge of him (1 Timothy 2:3-4), and the principles of salvation require all to be baptized and receive the Holy Ghost by the laying on of hands, then what of those who are incapable of meeting this requirement because they are no longer in possession of a physical body? There have been many people who lived their entire lives and died without ever so much as hearing about Jesus Christ, because of where they lived. Is it consistent with the idea of a loving God to place His children in such circumstances and then condemn them for it?

The doctrine of baptism for the dead by proxy (and also of the laying on of hands to confer the gift of the Holy Ghost, by proxy, which is part of the same process) is the resolution to this problem. It allows Latter-Day Saints who can obtain proof that a certain person existed, and that they lived and died without receiving the benefits of the Gospel, to receive the ordinances of salvation on their behalf, by proxy.

The doctrine is that this allows (but does not compel) the departed to accept the work that has been done for them, and progress in their understanding of Gospel principles, which are being taught to them (1 Peter 4:6), that they might gain salvation alongside those who were able to accept the Gospel in the flesh.

So in response to the question above, these ordinances are not performed for people who were Mormon, but for people who lived and died without accepting the restored Gospel. With regard to point #5, I'm not sure what the doctrine is on performing proxy work on behalf of people who chose to leave the church. If anyone does know, feel free to edit it in here, with a citation.