Everstone is an item that prevents a Pokémon from evolving when it is held. In the games, Everstones are useful as they remove the hassle of having to cancel the evolution of a Pokémon each time it levels up past the level where it evolves (like when you don't want a Pokémon with high friendship to evolve or meeting the right conditions). In the Generation IV games, it prevents evolutions during trades.
One should note it is unnecessary for Pokémon that evolve via Evolution Stones (like Pikachu) to hold one; however, it is useful for Pokémon, like Eevee or a female Snorunt that can evolve into two or more different forms.
This item can also be used in Breeding. If the mother (or Ditto) is holding an Everstone, the baby has a 50% chance of inheriting that Pokémon's Nature. It should be noted that Ditto always acts as the "mother" in a breeding match-up, so in Emerald, a female breeding with a Ditto does not have a better chance of passing down Natures to its offspring; the Ditto does.
In Generation IV, in Diamond, Pearl, and Platinum, inheriting Natures remains the same as in Emerald. However, in the Johto remakes, HeartGold and SoulSilver, inheriting Nature was changed a little. Whichever Pokémon is holding an Everstone will pass down its Nature, regardless of gender, or whether it is breeding with a Ditto. Another minor change that is present in all Generation IV games, is that the passing on of Natures via an Everstone will not work if the Pokémon are from different real-world countries (e.g. Argentina and Spain). This restriction is lifted in Generation V.
In Black 2 and White 2 the odds of passing down the Nature of the parent holding an Everstone is now 100%. If both parents are holding an Everstone, the odds are 50/50 between each of the parents' Natures.