The Ender Watcher is a neutral mob that spawns in The End. They can fill up the dimension quite quickly due to their prolific spawn rates. Ender Watchers will not attack a player unless provoked, or looked at, just like the Enderman. They will attempt to teleport behind a player to attack them, quite like Endermen. Ender Triplets will kill them if they are shooting at you when the Ender Watcher is provoked (if they're following you of course). Like every ender mob, projectile weapons will not hit the Ender Watcher.
Farming Method: While difficult to farm anything in The End with the open space for the Ender Triplets to attack, having a large room two blocks high is the perfect means to take on any mob (minus Enderman, since they require a space 3 blocks tall to spawn) in The End. Since Ender Watchers drop numerous ender shards, this is the best method to farm for Ender Armor and the Ender Sword. 2013-02-21 10.24.17 Spawn Room in The End As seen in the photo, a two high room below the top level of The End allows mobs to spawn without the worry of being attacked from overhead by Ender Triplets, though Ender Triplets spawn in a two high room as well, however since Ender Triplets have no room to fly, they are stuck in the position they spawn in. Refer to the Ender Triplets link above for more info on how to avoid/attack them in the spawn room. Of course, once you have a powerful sword that does more than 60 damage (like the Mythril Blade or the Halite Blade, preferably with Looting III) and armor sufficient to take Ender Triplet attacks, you can just run around The End and one-shot anything you see to get many more Ender Shards.
Behavior in Spawn Room: While every ender mob teleports when close by, the teleportation trigger on the Ender Watcher is on the antenna on the top of its head. To avoid it teleporting away while attacking it (if it teleports, it will teleport out of the room and will not teleport back in), attack the lower range of it or the back of its head.
Combat Tactics: As previously mentioned, the Ender Watcher likes to teleport behind the player. Take advantage of this by 1. Provoke an Ender Watcher. 2. Run in one direction 3. Turn around quickly and the Ender Watcher is most likely behind you. 4. Kill it and repeat.
If set on fire by tools or weapons such as the Inferno Sword, ender related mobs like the Ender Watcher will teleport rapidly due to the fire damage until they go far away and despawn.