In Pokémon Ruby and Sapphire only, due to a minor glitch, when collecting money dropped by Pay Day, the amount displayed will be incorrect if it was boosted by an Amulet Coin (the correct amount will still be obtained).
An Amulet Coin appeared in Capacia Island UFO!, where it was found by Ash. Later, when he and Team Rocket entered a UFO belonging to a group of Beheeyem, the Beheeyem told them that they lost an item looking like the Amulet Coin that was a vital part in getting the UFO to fly. Ash gave them the Amulet Coin he found, but it was not the right item. Instead, the Beheeyem sought a similar-looking item, which had been found by James earlier on.
In the anime, the Amulet Coin is regarded as a good luck charm.