I am rereading the book of Alma, and came across a chapter that in the past, never made much sense to me. In retrospect, I think it's because I had an incorrect understanding of the priesthood.
I now believe that the priesthood is simply a fellowship--Aaronic priesthood is a fellowship with angels, and Melchizedek priesthood is fellowship with God. Let's look at this definition of the Melchizedek priesthood from D&C 84:19-22: "And this greater priesthood administereth the gospel and holdeth the key of the mysteries of the kingdom, even the key of the knowledge of God. Therefore, in the ordinances thereof, the power of godliness is manifest. And without the ordinances thereof, and the authority of the priesthood, the power of godliness is not manifest unto men in the flesh; for without this no man can see the face of God, even the Father, and live."
(A side note--I'm writing these points as if I were sure about all of them. I am only writing that way because it's annoying to have to keep repeating "in my opinion" and "I think". What I write here represents my current understanding of the topic, and is subject to further development.)
So the "ordinances" are the things that enable us to stand in the presence of God (this ability being the "power of godliness"). These things include steps we must take to prepare, things Christ does for us to cleanse us, and probably the appearance itself.
"Authority" in the context of the priesthood does not imply power or control (as in the phrase, "to have authority over someone"); it means genuineness. Just as God's authority lies in his message itself because it is true, or in his actual delivery of his message (D&C 1:6), we would be able to bear authoritative testimony because we had actually witnessed God's appearing to us. It would be real, and that would be the source of our "authority."
We make a big deal of duties and responsibilities of priesthood holders in the Church and we confuse them with keys, but those responsibilities are mostly attached to offices, which are independent from the priesthood and mostly modern inventions based on organizational needs of today's Church. All the rules about which age groups may prepare, bless, and pass the sacrament is a good example of this, considering that it used to be the women who prepared the sacrament, and every Sunday, all members of the ward pass the sacrament to each other. We talk about keys, but never adequately explain them. I would argue that keys mean access to something, or a way to open a door or make a connection that wasn't there previously. If the door is not opened, it means nothing additional has been accessed, and therefore there is no authority. (It is useful to survey the use of the word "keys" in the D&C to identify which keys are mentioned in the scriptures and what they are for. Compare that list to what actually gets used in today's church.)
So this is how I have come to understand what Alma was trying to explain in chapter 13:
1. And again, my brethren, I would cite your minds forward to the time when the Lord God gave these commandments unto his children; and I would that ye should remember that the Lord God ordained priests, after his holy order, which was after the order of his Son, to teach these things unto the people.
So the order of God is "after" the order of his Son. Sequentially? Or patterned after? Probably the second. And what does "order" mean? The 1828 Webster's dictionary is a good resource here. It could mean "command", that God commanded things to be so; or it could mean "rank", as in God calling men up into the ranks with his Son; or it could mean "religious fraternity", as in joining the fellowship I mentioned.
2. And those priests were ordained after the order of his Son, in a manner that thereby the people might know in what manner to look forward to his Son for redemption.
So the way the ordination happens is important--it is meant to teach us how to look to Christ for redemption. How were these high priests ordained? Based on what I just read from D&C 84, I would suggest that they were high priests because they had stood in the presence of God, and had fellowship with Christ (receiving the ordinances, in which the power of godliness is manifest). Achieving that fellowship with Christ is the goal of redemption.
3. And this is the manner after which they were ordained--being called and prepared from the foundation of the world according to the foreknowledge of God, on account of their exceeding faith and good works; in the first place being left to choose good or evil; therefore they having chosen good, and exercising exceedingly great faith, are called with a holy calling, yea, with that holy calling which was prepared with, and according to, a preparatory redemption for such.
Question: what did God have foreknowledge about? That these particular individuals would turn out to be good enough for the task, and therefore they were the ones called? I would contend that the calling was foreordained, and the ordinances (preparatory steps) were predetermined, according to God's foreknowledge that we would need the help. In some circles, this is called the Plan of Salvation, and it is available to all of us.
4. And thus they have been called to this holy calling on account of their faith, while others would reject the Spirit of God on account of the hardness of their hearts and blindness of their minds, while, if it had not been for this they might have had as great privilege as their brethren.
This verse confirms that the foreknown calling was available to all, and whether a person received it depended on their faith and good works, and not on whether they were on some predetermined list of special people.
5. Or in fine, in the first place they were on the same standing with their brethren; thus this holy calling being prepared from the foundation of the world for such as would not harden their hearts, being in and through the atonement of the Only Begotten Son, who was prepared--
Can someone be saved through the atonement of Christ without being called? I suggest that the calling and being redeemed are synonymous, or at least simultaneous.
6. And thus being called by this holy calling, and ordained unto the high priesthood of the holy order of God, to teach his commandments unto the children of men, that they also might enter into his rest--
Once a person has received the calling, it is their job to teach others. Teach them what? To follow the very same path to Christ and fellowship with God.
7. This high priesthood being after the order of his Son, which order was from the foundation of the world; or in other words, being without beginning of days or end of years, being prepared from eternity to all eternity, according to his foreknowledge of all things--
8. Now they were ordained after this manner--being called with a holy calling, and ordained with a holy ordinance, and taking upon them the high priesthood of the holy order, which calling, and ordinance, and high priesthood, is without beginning or end--
9. Thus they become high priests forever, after the order of the Son, the Only Begotten of the Father, who is without beginning of days or end of years, who is full of grace, equity, and truth. And thus it is. Amen.
This salvation or calling is permanent, and this merely asserts that the ways and means of salvation have never changed, and never will--it is the same for everyone.
10. Now, as I said concerning the holy order, or this high priesthood, there were many who were ordained and became high priests of God; and it was on account of their exceeding faith and repentance, and their righteousness before God, they choosing to repent and work righteousness rather than to perish;
11. Therefore they were called after this holy order, and were sanctified, and their garments were washed white through the blood of the Lamb.
12. Now they, after being sanctified by the Holy Ghost, having their garments made white, being pure and spotless before God, could not look upon sin save it were with abhorrence; and there were many, exceedingly great many, who were made pure and entered into the rest of the Lord their God.
Reminding us of the basic steps of the Gospel that take us to Christ.
13. And now, my brethren, I would that ye should humble yourselves before God, and bring forth fruit meet for repentance, that ye may also enter into that rest.
A direct invitation to us. And finally, a specific example of how this process worked successfully in the past:
14. Yea, humble yourselves even as the people in the days of Melchizedek, who was also a high priest after this same order which I have spoken, who also took upon him the high priesthood forever.
15. And it was this same Melchizedek to whom Abraham paid tithes; yea, even our father Abraham paid tithes of one-tenth part of all he possessed.
16. Now these ordinances were given after this manner, that thereby the people might look forward on the Son of God, it being a type of his order, or it being his order, and this that they might look forward to him for a remission of their sins, that they might enter into the rest of the Lord.
17. Now this Melchizedek was a king over the land of Salem; and his people had waxed strong in iniquity and abomination; yea, they had all gone astray; they were full of all manner of wickedness;
18. But Melchizedek having exercised mighty faith, and received the office of the high priesthood according to the holy order of God, did preach repentance unto his people. And behold, they did repent; and Melchizedek did establish peace in the land in his days; therefore he was called the prince of peace, for he was the king of Salem; and he did reign under his father.
19. Now, there were many before him, and also there were many afterwards, but none were greater; therefore, of him they have more particularly made mention.
20. Now I need not rehearse the matter; what I have said may suffice. Behold, the scriptures are before you; if ye will wrest them it shall be to your own destruction.
To recap, these priests are not simply special people that the rest of us should look up to because God knew beforehand they would be successful in their quest and we wouldn't be (or even if he did know, that's not what made them special). They are people who rose up and accepted the universal invitation to come unto Christ, and because they did so, their lives bore authoritative witness that it is possible to be saved. It's up to us whether we join their fellowship.