Foreshadowing the ministry of Jesus Christ (Hebrews 2:17, 4:14 to 7:28, 9, 10), the High Priest represented all of God's people before the Eternal. His duties were the same as any priest with a few exceptions. He alone was to perform the tasks required on the Day of Atonement (Yom Kippur), which included entering the Temple's Holy of Holies (the only time he was permitted to do so each year) in order to offer blood for the people's sins (Leviticus 16). He alone was allowed to offer the regular meal offering (Leviticus 6:14 - 15).
The High Priest was also the only person who could consult the Urim and Thummim regarding a matter that required a decision directly from God (Exodus 28:30). In addition, he alone was allowed to wear a special breastplate (sometimes referred to as the "breastplate of judgment" - Exodus 28:30) that had embedded in it twelve precious stones which represented the twelve tribes of Israel (Leviticus 8:7 - 9, Exodus 28:17 - 20, 39:10 - 13).
The main purpose of this piece of clothing, with its names of Israel's tribes on stones embedded in it, was to be a memorial (reminder) before the Eternal. It was meant to remind the priest that he represented ALL the people, as mediator and intercessor, before the Lord (Exodus 28:29, 39:7).
The breastplate itself was to have the precious stones embedded in it placed in four rows containing three stones each (Exodus 28:17, 39:10). The name of an Israelite tribe was to be engraved on each stone (Exodus 28:21, 39:14).
Scripture does not directly state in what order the names were placed on the stones. However, first century Jewish Historian Josephus (who came from a priestly family) states that the names were placed in birth order (Antiquities of the Jews, Book 3, Chapter 7, Verse 5). This makes a certain amount of sense, since the Bible DOES state that the tribes were to be listed, six names each in birth order, upon two onyx stones the High Priest wore on his shoulders (Exodus 28:9 - 10).
The birth order of Jacob's (Israel's) male children, from firstborn to last, is Reuben, Simeon, Levi, Judah, Dan, Naphtali, Gad, Asher, Issachar, Zebulun, Joseph and Benjamin (Genesis 29 - 30).
Name and Gem order
The method most people in the Western world (including those who wrote the KJV, NKJV and most other Bibles) read and write is left to right (the method you are using to read this text). Ancient Hebrew, however, wrote and read starting on the right and going left. This means that the engraving of the precious stones, in the birth order of Jacob's son, begins on the far RIGHT of the breastplate as opposed to starting on the far left. For example, Reuben, the firstborn son of Jacob, should have his name engraved on the FAR RIGHT stone in the first row. Going right to left, Simeon's name is on the middle stone, while Levi's name is on the row's far left stone (there are three stones in each row). Judah, the fourth son, has his name on the far right stone in the second row, and so on (Gemstones in the Breastplate, page 7).
Hebrew's right to left method of reading and writing also affects the exact placement of stones within the breastplate. For example, based on this study series, the first three stones in the first row (Exodus 28:17), using our English method for listing things left to right, are the Carnelian, Peridot and an Emerald. The actual placement of the stones, however, begins with the first gemstone (Carnelian) placed on the FAR RIGHT of the first row. The middle (or second) stone, going right to left, is the Peridot, with the far left stone in the row being an Emerald.
Putting it all together, the below image shows the placement of precious stones in the High Priest's breastplate (from the point of view of a person looking at it while the priest is wearing it) along with the name engraved on each one.