An Everstone is a unique item. You will need to place it on a Pokémon that you want to prevent from evolving. You can also prevent a Pokémon from evolving by spamming the B Button while it is evolving into its next form, but this can get boring. Everstone does this automatically. Here's what you need to do to find an immobile Stone in Pokémon Brilliant Diamond and Brilliant Pearl.
The best way to find an Everstone is to explore the Grand Underground. You can reach this location once you receive the Explorer Kit from the old man from Echoes, the one who lives near the Pokémon Center. While in the Grand Underground, you will need to dig into the orange sites on the map. These are research sites where you can find rare items, such as Everstone.
You can increase your chances of finding items like Everstone by collecting Diglett and Dugtrio that appear on the ground. You will see the bonus steadily rise on the left side of the screen, below the Grand Underground map. The higher it is, the more chance you have of finding rare items. When you hit 40, you'll have a better chance of finding items, glowing statues, and Shiny Pokémon in the area for four minutes.