Verses of Exodus 28:23-30 discuss the mounting hardware and procedure of the Breastplate of Judgement to the Ephod (the Vestment). Let's take a look:
23 וְעָשִׂ֙יתָ֙ עַל־הַחֹ֔שֶׁן שְׁתֵּ֖י טַבְּע֣וֹת זָהָ֑ב וְנָתַתָּ֗ אֶת־שְׁתֵּי֙ הַטַּבָּע֔וֹת עַל־שְׁנֵ֖י קְצ֥וֹת הַחֹֽשֶׁן׃24 וְנָתַתָּ֗ה אֶת־שְׁתֵּי֙ עֲבֹתֹ֣ת הַזָּהָ֔ב עַל־שְׁתֵּ֖י הַטַּבָּעֹ֑ת אֶל־קְצ֖וֹת הַחֹֽשֶׁן׃25 וְאֵ֨ת שְׁתֵּ֤י קְצוֹת֙ שְׁתֵּ֣י הָעֲבֹתֹ֔ת תִּתֵּ֖ן עַל־שְׁתֵּ֣י הַֽמִּשְׁבְּצ֑וֹת וְנָתַתָּ֛ה עַל־כִּתְפ֥וֹת הָאֵפֹ֖ד אֶל־מ֥וּל פָּנָֽיו׃26 וְעָשִׂ֗יתָ שְׁתֵּי֙ טַבְּע֣וֹת זָהָ֔ב וְשַׂמְתָּ֣ אֹתָ֔ם עַל־שְׁנֵ֖י קְצ֣וֹת הַחֹ֑שֶׁן עַל־שְׂפָת֕וֹ אֲשֶׁ֛ר אֶל־עֵ֥בֶר הָאֵפֹ֖ד בָּֽיְתָה׃27 וְעָשִׂיתָ֮ שְׁתֵּ֣י טַבְּע֣וֹת זָהָב֒ וְנָתַתָּ֣ה אֹתָ֡ם עַל־שְׁתֵּי֩ כִתְפ֨וֹת הָאֵפ֤וֹד מִלְּמַ֙טָּה֙ מִמּ֣וּל פָּנָ֔יו לְעֻמַּ֖ת מֶחְבַּרְתּ֑וֹ מִמַּ֕עַל לְחֵ֖שֶׁב הָאֵפֽוֹד׃28 וְיִרְכְּס֣וּ אֶת־הַ֠חֹשֶׁן מִֽטַּבְּעֹתָ֞יו אֶל־טַבְּעֹ֤ת הָאֵפֹד֙ בִּפְתִ֣יל תְּכֵ֔לֶת לִֽהְי֖וֹת עַל־חֵ֣שֶׁב הָאֵפ֑וֹד וְלֹֽא־יִזַּ֣ח הַחֹ֔שֶׁן מֵעַ֖ל הָאֵפֽוֹד׃29 וְנָשָׂ֣א אַ֠הֲרֹן אֶת־שְׁמ֨וֹת בְּנֵֽי־יִשְׂרָאֵ֜ל בְּחֹ֧שֶׁן הַמִּשְׁפָּ֛ט עַל־לִבּ֖וֹ בְּבֹא֣וֹ אֶל־הַקֹּ֑דֶשׁ לְזִכָּרֹ֥ן לִפְנֵֽי־יְהוָ֖ה תָּמִֽיד׃30 וְנָתַתָּ֞ אֶל־חֹ֣שֶׁן הַמִּשְׁפָּ֗ט אֶת־הָאוּרִים֙ וְאֶת־הַתֻּמִּ֔ים וְהָיוּ֙ עַל־לֵ֣ב אַהֲרֹ֔ן בְּבֹא֖וֹ לִפְנֵ֣י יְהוָ֑ה וְנָשָׂ֣א אַ֠הֲרֹן אֶת־מִשְׁפַּ֨ט בְּנֵי־יִשְׂרָאֵ֧ל עַל־לִבּ֛וֹ לִפְנֵ֥י יְהוָ֖ה תָּמִֽיד׃ ס
23 And thou shalt make upon the breastplate two rings of gold, and shalt put the two rings on the two ends of the breastplate.
24 And thou shalt put the two wreathen chains of gold in the two rings which are on the ends of the breastplate.
25 And the other two ends of the two wreathen chains thou shalt fasten in the two ouches, and put them on the shoulderpieces of the ephod before it.
26 And thou shalt make two rings of gold, and thou shalt put them upon the two ends of the breastplate in the border thereof, which is in the side of the ephod inward.
27 And two other rings of gold thou shalt make, and shalt put them on the two sides of the ephod underneath, toward the forepart thereof, over against the other coupling thereof, above the curious girdle of the ephod.
28 And they shall bind the breastplate by the rings thereof unto the rings of the ephod with a lace of blue, that it may be above the curious girdle of the ephod, and that the breastplate be not loosed from the ephod.
29 And Aaron shall bear the names of the children of Israel in the breastplate of judgment upon his heart, when he goeth in unto the holy place, for a memorial before the LORD continually.
30 And thou shalt put in the breastplate of judgment the Urim and the Thummim; and they shall be upon Aaron's heart, when he goeth in before the LORD: and Aaron shall bear the judgment of the children of Israel upon his heart before the LORD continually.
Let's begin with our unmounted Breastplate of Judgement:
As Exodus 28:23 states, there were to be two rings of gold made at the top corners of the Breastplate, like so:
Exodus 28:24 states that to those two golden rings were to be attached two golden chains (as I have discussed in this post). Like so:
Exodus 28:25 states that on the other ends of these golden chains were to be attached special "mountings" or rather a sort of "hooks" that would allow to attach the Breastplate to the stones of the Ephod. Here are those mountings:
And here is the rendering of how these mountings and chains would be attached to the two "onyx" stones of the Ephod:
Exodus 28:26 states that there were to be made two more golden rings that would be attached to the bottom corners of the Breastplate. Like so:
Exodus 28:27 tells us that there were to be two more golden rings that would be attached to the actual Ephod (not the Breastplate) and those rings would be located right underneath the points where the golden mountings with chains would be attached to the stones of the Ephod. This is rather important detail, as it means that the mounting points of the Breastplate were located symmetrically so that the Breastplate would fit tightly onto the chest of the priest and would not move. Here is the rendering of those two Ephod rings from the top (Ephod is not shown for the illustration purposes):
Exodus 28:28 states that two bottom golden rings of the Breastplate were to be attached to the corresponding golden rings of the Ephod via two blue laces. Like so:
Thus, as Exodus 28:29 states, the priest (Aaron) would be able to bear the Breastplate on his chest without having to worry that it would move off or fall off, once the Breastplate was secured to the Ephod according to all the description above. Like so:
Last verse of the description (Exodus 28:30) describes the so-called "Urim" and the "Thummim" that were to be given to the Breastplate. I'm not sure what exactly does this mean as these two words can be translated in a different ways.
Here is the translation of this verse from my Normalized Interlinear Pentateuch:
Exodus 28:30: and_you-give to breastplate the_sentence with the_lights and_with the_perfections and_they-became over heart another in_coming_him to_faces_me yahweh and_lift another with sentence created_me prevailing-strength over heart_him to_faces_me yahweh continually.
So most likely, this verse means that by creating and mounting the Breastplate according to all specifications listed in the description, the Breastplate would become filled with "lights" (Urim) and "perfections" (Thummim). Meaning that it would become Holy and start to posses necessary qualities that were needed for the service of the Tabernacle. This is of'course is just my best guess at the moment, because further research is needed to resolve this question.
And here is final rendering of the mounted Breastplate of Judgement upon the Ephod (The Vestment) of the Priest: