A couple of items deserve mention. First, the phrase, "laying on [of] hands" is used only once in the Book of Mormon and it is found in Alma 6:1. It is interesting that this phrase is not used to describe the confirmation ordinance (see commentary for Mosiah 18:14). That it was used to ordain priests and elders is in accord with modern practices because it is according to the order of God in all dispensations.
Second, although it is clear that the Nephites had the Melchizedek priesthood, the offices in that priesthood were not structured exactly as they have been since the days of Christ. Rather the Book of Mormon speaks of four priesthood offices: high priest, elder, priest, and teacher (see commentary for Moroni 3:1). At this time, Alma is the only high priest (there is a similar pattern in the Levitical Priesthood under the Law of Moses). Alma's calling, then, was similar to what we know as the President of the Church, or the President of the High Priesthood of the Church (DC 107:65). Therefore, the term, "priests," as used in this verse has reference to an office in the Melchizedek Priesthood. Elder McConkie compared the Nephite office of "priest" in the Melchizedek Priesthood to the latter day calling of "high priest" in the Melchizedek Priesthood.
Bruce R. McConkie
"Book of Mormon prophets gave the title priest to officers known in this dispensation as high priests. That is, they were priests of the Melchizedek Priesthood, or as Alma expressed it, 'the Lord God ordained priests, after his holy order, which was after the order of his Son.' (Alma 13:1-20.) Since there was no Aaronic Priesthood among the Nephites in Alma's day (there being none of the lineage empowered in pre-meridian times to hold that priesthood), there was no need to distinguish between priests of the lesser and greater priesthoods." (Mormon Doctrine, p. 599)