As Joseph and Oliver worked on the translation of the Book of Mormon, they came to this passage in 3 Nephi 11:
21 And the Lord said unto him: I give unto you power that ye shall baptize this people when I am again ascended into heaven.
22 And again the Lord called others, and said unto them likewise; and he gave unto them power to baptize. And he said unto them: On this wise shall ye baptize; and there shall be no disputations among you.
23 Verily I say unto you, that whoso repenteth of his sins through your words, and desireth to be baptized in my name, on this wise shall ye baptize them—Behold, ye shall go down and stand in the water, and in my name shall ye baptize them.
24 And now behold, these are the words which ye shall say, calling them by name, saying:
25 Having authority given me of Jesus Christ, I baptize you in the name of the Father, and of the Son, and of the Holy Ghost. Amen.
26 And then shall ye immerse them in the water, and come forth again out of the water.
It was apparent from this text that the Savior gave the people power and authorized them to baptize. In discussing the matter, they reflected on how they, so far as they were aware, did not have such power. How then were baptisms to be done in this day and age? What happened next is recounted in the JS-H (the prayer is now D&C 13):
68 We still continued the work of translation, when, in the ensuing month (May, 1829), we on a certain day went into the woods to pray and inquire of the Lord respecting baptism for the remission of sins, that we found mentioned in the translation of the plates. While we were thus employed, praying and calling upon the Lord, a messenger from heaven descended in a cloud of light, and having laid his hands upon us, he ordained us, saying:
69 Upon you my fellow servants, in the name of Messiah, I confer the Priesthood of Aaron, which holds the keys of the ministering of angels, and of the gospel of repentance, and of baptism by immersion for the remission of sins; and this shall never be taken again from the earth until the sons of Levi do offer again an offering unto the Lord in righteousness.
70 He said this Aaronic Priesthood had not the power of laying on hands for the gift of the Holy Ghost, but that this should be conferred on us hereafter; and 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.
71 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.*
72 The messenger who visited us on this occasion and conferred this Priesthood upon us, said that his name was John, the same that is called John the Baptist in the New Testament, and that he acted under the direction of Peter, James and John, who held the keys of the Priesthood of Melchizedek, which Priesthood, he said, would in due time be conferred on us, and that I should be called the first Elder of the Church, and he (Oliver Cowdery) the second. It was on the fifteenth day of May, 1829, that we were ordained under the hand of this messenger, and baptized.
In 1999 I was asked to speak at the very first FAIR conference in Ben Lomond, California, which is the genesis of my involvement with that organization. In my presentation I compared and contrasted the restorations of Joseph Smith and Alexander Campbell. It’s a long paper, but the following sentence near the end will serve as a summary: “Similarities exist in the understanding of an apostasy resulting in the need for a restoration; the rejection of creeds, and in particular the rejection of metaphysical speculation regarding the Trinity; and in the conversion process of faith, repentance and baptism. Strong differences include variant understandings of the roles of religious authority, revelation from God to man, and the Old Testament.”
The restoration of the Aaronic Priesthood strikes me as an example of the kind of restoration that would have made no sense whatsoever to Alexander, given the Old Testament focus of that priesthood. But Joseph was of a class of seeker that sought religious authority. The importance of religous authority for these seekers can perhaps be grasped by this quotation from Parley P. Pratt:
About this time one Mr. Sidney Rigdon came into the neighborhood as a preacher, and it was rumored that he was a kind of Reformed Baptist, who, with Mr. Alexander Campbell, of Virginia, a Mr. Scott, and some other gifted men, had dissented from the regular Baptists, from whom they differed much in doctrine. At length I went to hear him, and what was my astonishment when I found he preached faith in Jesus Christ, repentance towards God, and baptism for remission of sins, with the promise of the gift of the Holy Ghost to all who would come forward, with all their hearts, and obey the doctrine!
Here was the ancient gospel in due form. Here were the very principles which I had discovered years before; but could find no one to minister in. But still one great link was wanting to complete the chain of the ancient order of things; and that was, the authority to minister in holy things–the apostleship, the power which should accompany the form. This thought occurred to me as soon as I heard Mr. Rigdon make proclamation of the gospel.
Peter proclaimed this gospel, and baptized for remission of sins, and promised the gift of the Holy Ghost, because he was commissioned so to do by a crucified and risen Saviour. But who is Mr. Rigdon? Who is Mr. Campbell? Who commissioned them? Who baptized them for remission of sins? Who ordained them to stand up as Peter? Of course they were baptized by the Baptists, and ordained by them, and yet they had now left them because they did not administer the true gospel. And it was plain that the Baptists could not claim the apostolic office by succession, in a regular, unbroken chain from the Apostles of old, preserving the gospel in its purity, and the ordinances unchanged, from the very fact that they were now living in the perversion of some, and the entire neglect of others of these ordinances; this being the very ground of difference between the old Baptists and these Reformers.
Again, these Reformers claimed no new commission by revelation, or vision from the Lord, while they had not the least shadow of claim by succession. It might be said then, with propriety: “Peter I know, and Paul I know, but who are ye?”
However, we were thankful for even the forms of truth, as none could claim the power, and authority, and gifts of the Holy Ghost–at least so far as we knew.
When we celebrate the restoration of the priesthood, we tend to focus on the Aaronic Priesthood, because we have a date certain for that event (May 15, 1829). (I have attended many ward campouts on or about that date.) We don’t know for sure when the Melchizedek Priesthood was restored. This old post by BCC perma Ben P. goes over the evidence for the two principal theories. (I once gave a talk based on Larry Porter’s Ensign article articulating the May/June 1829 theory, but these days I tend to lean to the minority view of Quinn and Bushman looking to July 1830.)
An important resource some instructors will find useful is Priesthood Restoration Documents, originally published in BYU Studies and later republished in Opening the Heavens.
Here is a wonderful, enlightening article that everyone needs to read: William G. Hartley, “From Men to Boys: LDS Aaronic Priesthood Offices, 1829-1996” Journal of Mormon History 22/1 (1996): 80-136. If you’ve ever had anyone ask how in the world a 16-year old kid can be a priest, this is the article for you.
One final thought: On our BCC backlist, some of the sistern have complained that they don’t really know much about the priesthood or how it works, because they were never taught about it as young women. So another angle to this lesson might be to engage the class in a primer on the basics of priesthood organization, and how things work in day to day practice. (The manual has a chart that might be useful toward this end.)
 Parley P. Pratt, Autobiography of Parley P. Pratt (Salt Lake City: Deseret Book Company, 1973), 31-32.