How does the judge function work?

At a certain point in the game, you can unlock the judge function, to see the potential of your Pokémon. I assume this are the IV stats within this game, opposed to the the AV already explained. What are the different textual values representing, in both upper corners?

This page is indeed a replica of the similar looking screen introduced in Sun and Moon to display IVs.

The text on the right describes how good the particular stats are:

• 31 - Best
• 30 - Fantastic
• 26-29 - Very Good
• 16-25 - Pretty Good
• 1-15 - Decent
• 0 - No Good

And the text on the left is for the overall total:

• 151-186 - Amazing Stats
• 121-150 - Great Stats
• 91-120 - Good Stats
• 0-90 - OK Stats

You can unlock this judge by talking to Professor Oak's Aide in the gate between routes 11 and 12.

You don't see it in yours, but usually you also get one stat name highlighted in blue and another in red -- this indicates the Pokémon's nature, with red being the raised stat and blue being the lowered stat (both by 10%). If your nature raises and lowers your same stat (also known as a "neutral nature"), none of them will be colored.

Source: Serebii