Arm-Mimics are recurring enemies in The Legend of Zelda series. They move by mirroring Link in opposite directions, in the same way as Shy Guys. If Link moves up, they will move down. If Link moves left, they will move right. A sword slash will do nothing but push the Arm-Mimic back. To defeat them, use spin attacks or charge through them with the Sword and Pegasus Boots.
In Link's Awakening, Arm-Mimics are found exclusively within Dream Shrine, Turtle Rock and the cave where Link obtains the Magnifying Lens. Arm-Mimics found in this cave are invisible and can only be seen with the Magnifying Lens. It is easiest to defeat them by using a Spin Attack or Charge Attack making it easier to defeat these enemies. Arm-Mimics deal three hearts of damage, making them the strongest enemies in the game.
Oracle of Seasons and Oracle of Ages
The Arm-Mimics in Oracle of Seasons and Oracle of Ages look and act exactly the same as in Link's Awakening. The differences are, they can withstand more damage, they cause only one heart damage, and they can be defeated with a normal sword slash. They are found in various dungeons.
To defeat them, either back them into a wall, or move carefully so that they will slowly move towards Link, and slash away at them a few times with the sword.
Orange Arm-Mimic sprite from Link's Awakening DX