Eater of Worlds

The Eater of Worlds is a Boss monster summoned by using Worm Food in The Corruption (manually planted corruption only counts if there are at least 300 ebonstone) as well as being summoned each time you break 3 Shadow Orbs.

Segments

 * 1-3 Shadow Scales: 100% (1:1 chance)
 * 2-5 Demonite Ores: 50% (1:2 chance)

Final Kill

 * 30-59 Demonite Ores: 100% (1:1 chance)
 * 5-15 Lesser Healing Potions: 100% (1:1 chance)
 * 1 Piece of Shadow Armor: 100% (1:1 chance)
 * Shadow Helmet: 33.33% (1:3 chance)
 * Shadow Scalemail: 33.33% (1:3 chance)
 * Shadow Greaves: 33.33% (1:3 chance)

Boss Description
Like all Worms, the Eater of Worlds is not a single target; Each body section is an individual entity. While destroying a single section is enough to kill every other Worm, this will not kill the Eater of Worlds. In fact, it will split the Eater of Worlds into several different sub-Worms. To defeat the Eater of Worlds, you must completely destroy each and every section of the Worm, until there is nothing left.

Each individual worm is composed of at least 2 entities: a Head, a Tail, or more Body sections. Splitting the Eater of Worlds into parts of less than 2 sections will instantly destroy the stray parts.

Each section of the Eater of Worlds is not equivalent: The Head of the Worm will deal significantly more damage than the Body and Tail. Direct contact with Heads should be avoided at all cost (Even if you can't fully avoid a Worm, a Body contact is better than a Head contact). The Head of the Eater of Worlds has significantly less Health than the body; This can be exploited when trying to hunt down the last small Worm.

Terrain and Gear
Like all Bosses preparing for the fight is probably the biggest factor in your success:

Standing on a wide, elevated platform (with either glass or sky below it) helps tremendously, as you can see him approaching from below. Further more, since Worms cannot navigate in the air, it will also drastically reduce the Eater of World's mobility. Under such a scenario, the Eater of Worlds can almost always be evaded: Getting hit is often a result of the player's own risk-taking to deal extra damage.

The Platform can further be upgraded with several levels of Wood Platforms, to further increase the player's mobility. Rocket Boots can also help, but are not a requirement.

Good Armor is not a strict necessity, as the Eater of Worlds tends to hit hard, rather than often.

Weapons
Because of its large size and multiple body parts, weapons that can deal damage to multiple sections should be privileged:Vilethorn, Ball 'O Hurt, Shurikens, Vile Powder, Space Gun, etc... Guns can also be very effective at killing Worms of small size.

Note that while weapons like the Ball 'O Hurt tend to damage the Eater of Worlds faster, it will also deal more focused damage, splitting the Eater of Worlds more often. The Ball 'O Hurt is excellent for killing the sub-Worms.

Vile Powder, on the other hand, will not deal its damage as fast (but still very fast), but will have a balanced damage distribution throughout the entire body. Vile Powder has the advantage of being easy to find and free. Furthermore, it is a very safe weapon to use against the worm: It can be placed before dodging the Eater of Worlds and running away, but it will still take damage. It is recommended to have 30 to 150 Vile Powders for this strategy, depending on how conservative you are, and how soon you make the switch from "Damage the Worm" to "Hunt the sub-Worms"

The choice of the weapon will essentially boil down to how safe you want play, or how fast you want to kill the Eater of Worlds.

Using dynamite will instantly kill any parts of the worm hit, providing an extremely fast (but expensive) way to kill the Eater of worlds.

Managing the Worms
Killing a section of the Eater of Worlds will split into an additional Worms with the Head of the new worm starting where you destroyed the segment. Of course, the more worms that are out, the harder he is. A fundamental strategy when killing the Eater of Worlds is correct management of each sub-Worm. Avoid splitting the Worm when you can. If there are little Worms, concentrate on killing those first, before returning to the main Worm.

Death
Failing to kill the Eater of Worlds will result in the creature chasing you back to spawn, with some delay depending on how far away you were when you were fighting him. He will relentlessly attack until you slay him. After some time he will disappear if you keep dying (an "Eater of Worlds is Defeated!" message will appear with no loot dropped). Occasionally, only one section will follow you if you die, and you can obtain a piece of Shadow Armor relatively easily.

"Surfing" the Worm
When using the above Platform scenario, and with proper movement, you can get the Eater of Worlds to chase you horizontally, as if it was surfing on your platform. This can be very useful once you are in the "Kill" phase of the fight, and to concentrate your fire on the Worms' Heads, which have less health than the Body. Using this strategy, entire sections of worms can be killed at once. The Ball 'O Hurt, Shurikens, or even a simple Gun, are all very effective.

Note that it is much easier to get the Worm to surf by having a two Block wide platform.

Video of how to beat the Eater of Worlds:

Fastest method to kill the eater of worlds and useful advice

Trivia

 * In update 1.0.4 the Eater of Worlds was updated to have slightly more HP and Defense and deal more damage.