17 And thou shalt set in it settings of stones, even four rows of stones: the first row shall be a sardius, a topaz, and a carbuncle: this shall be the first row.
18 And the second row shall be an emerald, a sapphire, and a diamond.
19 And the third row a ligure, an agate, and an amethyst.
20 And the fourth row a beryl, and an onyx, and a jasper: they shall be set in gold in their inclosings.
21 And the stones shall be with the names of the children of Israel, twelve, according to their names, like the engravings of a signet; every one with his name shall they be according to the twelve tribes.
The face of the breastplate would be covered with twelve precious stones. They would be four rows of three stones, and each would be inscribed with the name of one of the twelve tribes of Israel. We were already told that each priest would bear the twelve names upon his shoulders, six on each side engraved in a single onyx stone, and now all of them again on the chest, but here each carved into its own unique jewel.
There may be some symbolism in this. Having a cluster of names on each shoulder could mean having a broad sense of the burden for the work that the priest must perform for each tribe. The work for the Danite would be indistinguishable from the work for a Benjaminite. Each would be part of the same mass of labor, and so, too, their names could be massed together, six on each shoulder. But the tribes were not to be lumped together in such a way upon the heart. Twelve unique stones on the chest suggests having a unique and individual love for each tribe, the priest being able to appreciate their separate roles within the whole.
As for the types of stone that are described, some of them are assumed my modern scholars to be incorrectly interpreted. For example, there is no evidence that these ancient people would have known of the diamond, nor that they would have had the technology to etch a name into it. It therefore seems highly likely that a different stone was meant.
I have to assume that each stone was not chosen at random. If a sardius was selected for Reuben, I assume there was something in the nature of the sardius that was seen as particularly fitting for the nature, role, and destiny of Reuben within ancient Israel. There are several commentaries that suggest what those connections might have been, but I do not have the background to know which ones are most credible, so for now I will content myself with the opinion that the connections do exist.