I think there are a lot of misconceptions regarding the baptism by fire. I also think many think they have had this experience, when in reality they have not and ought to be seeking to have it. I figured a good place to begin an exploration of this doctrine is the scriptures.
Nephi describes this experience in 2 Nephi 31:7
Wherefore, do the things which I have told you I have seen that your Lord and your Redeemer should do; for, for this cause have they been shown unto me, that ye might know the gate by which ye should enter. For the gate by which ye should enter is repentance and baptism by water; and then cometh a remission of your sins by fire and by the Holy Ghost.He refers to the baptism by fire as "a remission of your sins by fire and by the Holy Ghost." This is the same as justification, being free from blame, being found guiltless, or having all your sins forgiven.
There are several clear examples of people who have had the baptism by fire.
 The Pentecost
After Christ's resurrection, he appeared to the Apostles and some others.
Jesus tells them to "Receive the Holy Ghost":
21 Then said Jesus to them again, Peace be unto you: as my Father hath sent me, even so send I you.22 And when he had said this, he breathed on them, and saith unto them, Receive ye the Holy Ghost:23 Whose soever sins ye remit, they are remitted unto them; and whose soever sins ye retain, they are retained. (John 20:21-23)Nothing spectacular happened then. There is no mention of the Spirit descending upon them. They continued to wait for that experience. It is evident from a reading of that chapter and Acts 1 that they didn't yet receive the Holy Ghost.
4 And, being assembled together with them, commanded them that they should not depart from Jerusalem, but wait for the promise of the Father, which, saith he, ye have heard of me.5 For John truly baptized with water; but ye shall be baptized with the Holy Ghost not many days hence. (Acts 1:4-5)So they waited in Jerusalem and on the day of Pentecost, they finally had the experience they had been waiting for.
1 And when the day of Pentecost was fully come, they were all with one accord in one place.2 And suddenly there came a sound from heaven as of a rushing mighty wind, and it filled all the house where they were sitting.3 And there appeared unto them cloven tongues like as of fire, and it sat upon each of them.4 And they were all filled with the Holy Ghost, and began to speak with other tongues, as the Spirit gave them utterance. (Acts 2:1-4)A few things happened here: They heard the sound of wind, they saw fire, and they were all filled with the Holy Ghost and spiritual gifts were manifest.
 Cornelius' Family
Peter was told to preach to Cornelius and his family. After preaching, they had an amazing experience.
44 While Peter yet spake these words, the Holy Ghost fell on all them which heard the word.45 And they of the circumcision which believed were astonished, as many as came with Peter, because that on the Gentiles also was poured out the gift of the Holy Ghost.46 For they heard them speak with tongues, and magnify God. Then answered Peter,47 Can any man forbid water, that these should not be baptized, which have received the Holy Ghost as well as we?48 And he commanded them to be baptized in the name of the Lord. Then prayed they him to tarry certain days. (Acts 10:44-48)This is a very interesting case because these people were baptized by fire before being baptized by water. Spiritual gifts were also manifest.
 The Lamanites
Nephi and Lehi were preaching among the Nephites and were then imprisoned. There was an earthquake and darkness and Christ spoke. Aminadab, a Nephite dissenter told them to pray and repent. They prayed till the darkness left.
43 And it came to pass that when they cast their eyes about, and saw that the cloud of darkness was dispersed from overshadowing them, behold, they saw that they were encircled about, yea every soul, by a pillar of fire.44 And Nephi and Lehi were in the midst of them; yea, they were encircled about; yea, they were as if in the midst of a flaming fire, yet it did harm them not, neither did it take hold upon the walls of the prison; and they were filled with that joy which is unspeakable and full of glory.45 And behold, the Holy Spirit of God did come down from heaven, and did enter into their hearts, and they were filled as if with fire, and they could speak forth marvelous words.46 And it came to pass that there came a voice unto them, yea, a pleasant voice, as if it were a whisper, saying:47 Peace, peace be unto you, because of your faith in my Well Beloved, who was from the foundation of the world.48 And now, when they heard this they cast up their eyes as if to behold from whence the voice came; and behold, they saw the heavens open; and angels came down out of heaven and ministered unto them. (Helaman 5:43-48)In this experience fire encircled them, the Holy Ghost filled their hearts, spiritual gifts were manifested (speaking marvelous things), they heard a voice, and angels ministered to them. Something to ponder on is whether these Lamanites were previously baptized by water.
 Lorenzo Snow
This is taken from chapter 3 in the Teachings of the Presidents of the Church manual. The entire description of his experience is worth reading; however, here is a segment of it:
"I believed they had the true religion, and I joined the Church. So far my conversion was merely a matter of reason." He remembered, "I was perfectly satisfied that I had done what was wisdom for me to do under the circumstances." Although he was content for a time with this understanding, he soon yearned for a special manifestation of the Holy Ghost. He said, "I had had no manifestation, but I expected one." "This manifestation did not immediately follow my baptism, as I expected," he recalled. "But, although the time was deferred, when I did receive it, its realization was more perfect, tangible and miraculous than even my strongest hopes had led me to anticipate. It was a complete baptism—a tangible immersion in the heavenly principle or element, the Holy Ghost; and even more real and physical in its effects upon every part of my system than the immersion by water;"His baptism by fire did not come immediately following his baptism by water. His experience with the Holy Ghost felt like a complete immersion and even more real than his baptism by water.
 King Benjamin's People
King Benjamin gathered the people to the temple and gave a speech.
1 And now, it came to pass that when king Benjamin had made an end of speaking the words which had been delivered unto him by the angel of the Lord, that he cast his eyes round about on the multitude, and behold they had fallen to the earth, for the fear of the Lord had come upon them.2 And they had viewed themselves in their own carnal state, even less than the dust of the earth. And they all cried aloud with one voice, saying: O have mercy, and apply the atoning blood of Christ that we may receive forgiveness of our sins, and our hearts may be purified; for we believe in Jesus Christ, the Son of God, who created heaven and earth, and all things; who shall come down among the children of men.3 And it came to pass that after they had spoken these words the Spirit of the Lord came upon them, and they were filled with joy, having received a remission of their sins, and having peace of conscience, because of the exceeding faith which they had in Jesus Christ who should come, according to the words which king Benjamin had spoken unto them. (Mosiah 4:1-3)He then spoke some more about what it means to retain a remission of one's sins (which is an interesting thing to consider).
1 And now, it came to pass that when king Benjamin had thus spoken to his people, he sent among them, desiring to know of his people if they believed the words which he had spoken unto them.2 And they all cried with one voice, saying: Yea, we believe all the words which thou hast spoken unto us; and also, we know of their surety and truth, because of the Spirit of the Lord Omnipotent, which has wrought a mighty change in us, or in our hearts, that we have no more disposition to do evil, but to do good continually.3 And we, ourselves, also, through the infinite goodness of God, and the manifestations of his Spirit, have great views of that which is to come; and were it expedient, we could prophesy of all things.4 And it is the faith which we have had on the things which our king has spoken unto us that has brought us to this great knowledge, whereby we do rejoice with such exceedingly great joy. (Mosiah 5:1-4)They were led to have this experience because the Spirit showed them their carnal state. They felt really bad about their sins and wanted to better. They then prayed for the baptism by fire. With this experience, they had no desire to do wrong (their hearts were changed) and spiritual gifts were manifest (they could prophesy) and finally, they experienced great joy.
There are several other examples of people in the scriptures who were baptized by fire. These are worth reading about:
Adam (Moses 6:65)
The Nephite 12 (3 Nephi 19:13)
Jesus (Matthew 3:16)
Too many people erroneously believe that the baptism by fire is a process, rather than a point event. In every single case ever described in scripture, it is an event that is not on-going, but experienced in a moment.
If you have not had this experience, we can pray to have this experience. But, we do need to meet the Lord's requirement for it, which is contained in 3 Nephi 9:20.
And whoso cometh unto me with a broken heart and a contrite spirit, him will I baptize with fire and with the Holy Ghost.Although having a broken heart and a contrite spirit is a relative thing (since we can always become more contrite), there comes a point when it is acceptable and we are then baptized by fire.
Is it possible not to know you've had this experience? Consider this verse describing the Lamanites spoken of previously.
20 And ye shall offer for a sacrifice unto me a broken heart and a contrite spirit. And whoso cometh unto me with a broken heart and a contrite spirit, him will I baptize with fire and with the Holy Ghost, even as the Lamanites, because of their faith in me at the time of their conversion, were baptized with fire and with the Holy Ghost, and they knew it not. (3 Nephi 9:20)I think it is less likely that they didn't remember the experience and more likely that they didn't understand the experience and what was occurring at the time.
Let's summarize what we know about the experience:
- An experience had in conjunction with baptism by water
- Can occur before baptism by water
- Entails the remission of sins – you will know that your sins have all been forgiven
- Spiritual gifts are given
- Occurs as a point event (and is not a process)
- Requirements are a broken heart and a contrite spirit