Rebaptism

At this time came a revelation, that the Saints could be baptized and re-baptized when they chose. (Brigham Young J.D. 18:241)

Nearly All the church have been Baptized again, for the Remission of their Sins, since they joined the Church, I have also, by the hands of Br. Joseph (as he himself has been,) & I would advise Jan and you Mary, to attend to it as soon as you can have the opportunity of an Elder or Priest of the Church to administer it. (Jacob Scott from Nauvoo in February, 1843)

I want to address this. I had never heard of such a doctrine growing up as a LDS. I never realized it was at one time, a very common practice that has been forgotten. Like partaking of the sacrament, rebaptism was an opportunity to re-commit oneself to God.

Summary
Baptism was performed for the following reasons.

1. Entrance or re-entrance into the Church
2. Establishing dates for lost records
3. Renewing of covenants
4. Entering the United Order
5. Entering marriage
6. Accepting of Church positions as bishops, stake presidents, and the Apostleship
7. Healing of the sick
8. Remission of sins
9. And others, such as preparatory to missions, etc.
10. Renewal of the Command to be born again.

Alma 5:62 I speak by way of command unto you that belong to the church; and unto those who do not belong to the church I speak by way of invitation, saying: Come and be baptized unto repentance, that ye also may be partakers of the fruit of the tree of life.

Renewal of Covenants

“Brother Harris was taught the necessity of being re-baptized. He said that was new doctrine to him. Revelations 2nd Chapter was explained, that those who had lost their first love and had fallen into evils and snares, were called on to “repent and do their first works,” and that re-baptism was a part of the gospel. He claimed that he had not been cut off from the Church, but said if that was required of him it would be manifest to him by the Spirit. Soon after his arrival in Utah he applied for baptism, saying that the Spirit had made known to him that it was his duty to renew his covenant before the Lord.” (Life of Martin Harris, Millenial Star 44:87) 

...The proposition was joyfully and very generally accepted by the saints. This procedure, however, must not be regarded as casting any doubt upon the validity of their original baptisms, or repudiation of it as a sacrament. It was only to make more solemn the renewal of covenants' with God. (C.H.C., B. H. Roberts, 2:286-287) 

Healing.

Wed. 5. My dear Emma was worse. Many fears were entertained that she would not recover. She was baptized twice in the river, which evidently did her much good. She grew worse again at night, and continued very sick indeed. I was unwell, and much troubled on account of Emma's sickness. (History of Church, J. Smith, Vol. 5; p. 167-168)

Upon arriving at Pisgah, Lorenzo and members of his family remained for a season. "At this place," Lorenzo said, "I was taken seriously and dangerously ill with a burning fever, which so affected my brain that I was delirious many days, lying at the point of death. While in this condition, Elder Phineas Richards, the father of Apostle F. D. Richards, assisted by other kind brethren, took me from my bed, wrapped in a sheet placed me in a carriage, drove to a stream of water, and baptized me in the name of the Lord, for my recovery. The fever immediately abated, and through the kind unwearied nursing and attention by my faithful, loving wives, and my dear sister, E.R.S. Smith, aided and sanctified through the power and blessing of God, I was delivered from suffering and restored to health. The sickness was the result of extreme hardships and exposures consequent on the journey. (Life of L. Snow, 1846, Romney, p. 80-81)

Mrs. Richards was taken by her brother from a sick bed to a lake from the surface of which ice more than a foot thick [53] had been removed, and there baptized, whereupon she immediately recovered. Similar cases might be given by the score. (Bancroft's History of Utah, p. 337)

Remission of Sins

I was baptized first in 1832, and I was baptized the next time when I came out here to Utah. I was baptized first for the remission of sins. Was baptized the second time for the same thing. I came out here in 1847 and was re-baptized then. When I came out here I was baptized again renewing my covenants and also for the remission of sins that I might have committed in taking this long and tedious journey through all these mountains and canyons. (Joseph C. Kingsbury, Temple Lot Case, p. 340) 

My counsel to them today is, as it has been on former occasions to all who have [58] come into these valleys, Go and be baptized for the remission of sins, repenting of all your wanderings from the path of righteousness, believing firmly in the name of Jesus Christ, that all your sins will be washed away. If any of you inquire what is the necessity of your being baptized, as you have not committed any sins, I answer, it is necessary to fulfill all righteousness. I have heard some of you cursing and swearing, even some of the Elders of Israel. I would be baptized seven times, were I in your place; I would not stop teasing some good Elder to baptize me again and again, until I could think my sins forgiven. I would not live over another night until I was baptized enough to satisfy me that my sins were forgiven. Then go and be confirmed, as you were when you first embraced the religion of Jesus. That is my counsel. (Brigham Young, J.D. 2:8, 9) 

In the first place, if you were re-baptized for the remission of sins, peradventure you may receive again the Spirit of the Gospel in its glory, light and beauty; but if your hearts are so engrossed in the things of this world, that you do not know whether you want to be re-baptized or not, you had better shut yourselves up in some canyon or closet, to repent of your sins, and call upon the name of the Lord, until you get His spirit. (Brigham Young, J.D. 1:324)

Re-Entrance:

In the afternoon, I attended a council of the Twelve at President Young's. There were present: Brigham Young, Heber C. Kimball, Orson Hyde, Orson Pratt, John Taylor, Wilford Woodruff, George A. Smith, Willard Richards and Brother Hyrum Smith. . . . This Council was called to re-consider the case of Orson Pratt, who had previously been cut off from the quorum of the Twelve for the neglect of duty; and Amasa Lyman had been ordained an Apostle in his place. I told the Council that as there was not a quorum present when Orson Pratt's case came up before them, that he was still a member, that he had not been cut off legally, and I would find some other place for Amasa Lyman to which the Council agreed. President Young said there were but three present when Amasa was ordained. I told them that was legal when no more could be had. * * * At three o'clock, Council adjourned to my house, and at four I baptized Orson Pratt and his wife, Sarah Marinda, and Lidia Granger in the Mississippi River, and [62] confirmed them in the Church, ordaining Orson Pratt to his former office and standing in the Quorum of the Twelve. (History of Joseph Smith, Mill. Star, Vol. 20:423) 

Scriptures on Baptism:

And now I speak concerning baptism; Behold, elders, priests, and teachers were baptized; and they were not baptized save they brought forth fruit meet that they were worthy of it. (Moroni 6:1)

...baptism is a sign ordained of God, for the believer in Christ to take upon himself in order to enter into the kingdom of God, "for except ye are born of water and of the Spirit ye cannot enter into the Kingdom of God," said the Savior. It is a sign and a commandment which God has set for man to enter into His kingdom. Those who seek to enter in any other way will seek in vain; for God will not receive them, neither will the angels acknowledge their works as accepted, for they have not obeyed the ordinances, nor attended to the signs which God ordained for the salvation of man, to prepare him for, and give him a title to, a celestial glory; and God had decreed that all who will not obey His voice shall not escape the damnation of hell. What is the damnation of hell? To go with that society who have not obeyed His commands. (TPJS, p. 198)

Many objections are urged against the Latter-day Saints for not admitting the validity of sectarian baptism, and for withholding fellowship from sectarian churches. Yet to do otherwise would be like putting new wine into old bottles, and putting old wine into new bottles. What: New revelations in the old churches? New revelations would knock out the bottom of their bottomless pit. New wine into old bottles! The bottles burst and [9] the wine runs out: What! Sadducees in the new church! Old wine in new leathern bottles will leak through the pores and escape. So the Sadducee saints mock at authority, kick out of the traces, and run to the mountains of perdition, leaving the long echo of their braying behind them. (TPJS, p. 192)

Baptism is compared with dead works.

...although a man should be baptized an hundred times it availeth him nothing, for you cannot enter in at the strait gate by the law of Moses, neither by your dead works. (D. & C. 22:2)

When Jesus said we must be "born again," He illustrated baptism as a new way of life and new beginning. Acceptance of what He has begun anew among men.

Rebaptism: A Revelation.

“I know that in my traveling and preaching, many a time I have stopped by beautiful streams of clear, pure water, and have said to myself, “How delightful it would be to me to go into this, to be baptized for the remission of my sins.” When I got home, Joseph told me it was my privilege. At this time, came a revelation, that the Saints could be baptized and re-baptized when they chose, and then that we could be baptized for our dear friends.” (Brigham Young, J.D. 18:241) 

Scriptural Accounts of Rebaptism 
The Book of Mormon also contains specific references to re-baptism. Nephi, it is written, baptized many believers and "ordained men unto this ministry," and "there were many in the commencement of this year that were baptized unto repentance." (See 3 Nephi 7:26) However, a few years later, after the destruction of his nation, it is written that Nephi again "went down into the water and was baptized." (3 Nephi 19:11) Thus Nephi complied with the principle of re-baptism; and being a leader among his people, he established the precedent for that practice and they followed his example. Nephi proceeded to baptize the others and "it came to pass when they were ALL baptized and had come up out of the water, the Holy Ghost did fall upon them." (3 Nephi 19:13)

Alma also participated in the principle of re-baptism. After he received the authority and had been converted, then he preached the principles of faith, repentance, and baptism, and baptized a "goodly number" of the believers. Then on a later occasion, while baptizing a man by the name of Helam, he became so influenced with the beauty and blessings of baptism, that he also was "buried" in the waters of baptism. Alma, therefore, certainly experienced a re-baptism, and he "came forth out of the water rejoicing, being filled with the Spirit." (Mos. 18:13-14)

And now I speak concerning baptism. Behold, elders, priests, and teachers were baptized; and they were not baptized save they brought forth fruit meet that they were worthy of it. (Moroni 6:1)

The "elders, priests, and teachers" were of course members and offices of the Church of Jesus Christ, and yet they were baptized. To become members of the Church and hold the offices of elders and priests, they must be baptized; therefore, these members were again baptized after they had repented by bringing forth the "fruit" of repentance. Else one does not have to be baptized to receive the these offices spoken of in Moroni.

The Early Church

…he commanded us to go and be baptized, and gave us directions that I should baptize Oliver Cowdery, and that afterwards he should baptize me. Accordingly we went and were baptized. I baptized him first, and afterwards he baptized me—after which I laid my hands upon his head and ordained him to the Aaronic Priesthood, and afterwards he laid his hands on me and ordained me to the same Priesthood—for so we were commanded. (JS-H 1:70-71)

Later, in 1830, the church was officially organized and those first members were baptized. Among these people (nine in all) was Joseph Smith. The Desert News states:

…Joseph Smith and those who had been baptized prior to April 6, 1830, were again baptized on the day of the organization of the Church. (Deseret News, March 30, 1935, page 6.)

I attended meeting. Elder Erastus Snow preached, after which I addressed the congregation, and particularly the Elders, on the principles of wisdom, etc. President Rigdon preached in the afternoon, and several were confirmed among whom was Frederick G. Williams, who had recently been re-baptized. (Aug. 5, 1838, D.H.C. 3:55)

This is a representation of the feelings of some who have crossed the plains this season. My advice to you is, go and be baptized for the remission of sins, and start afresh, that temptation may not overcome you again; pause and reflect, that you be not overcome by the evil one unawares. (Brigham Young, J.D. 1:324)

In summary, it was practiced for the following reasons:

1. Entrance or re-entrance into the Church
2. Establishing dates for lost records
3. Renewing of covenants
4. Entering the United Order
5. Entering marriage
6. Accepting of Church positions as bishops, stake presidents, and the Apostleship
7. Healing of the sick
8. Remission of sins
9. And others, such as preparatory to missions, etc.
10. Renewal of the Command to be born again.

Further reading:
D Michael Quinn, "The Practice of Rebaptism at Nauvoo"
https://ojs.lib.byu.edu/spc/index.php/BYUStudies/article/viewFile/5015/4665

*Many quotes compiled by Ogden Kraut essay on Rebaptism.