Fighting the Brain of Cthulhu is very similar to fighting the Eye of Cthulhu, with a little tweak on phases. If you have the basics of how to Fight the Eye of Cthulhu, then you will have a better chance fighting the Brain.
Terrain Preparation
Standard 5 layer arena.
You'll want to set up a long platform in The Crimson, because it can only be summoned there. It can be summoned in caverns and above ground. It should be high enough so ground enemies can't jump up to you, but low enough that you don't accidentally go high enough to make it despawn. It is similar to the Eater of Worlds in the way that it can be summoned and fought during day or night. An arena the size of your screen is preferred, since you can see the whole area in the middle, it is safe, and enemies can't go jumping in. It is suggested to place a Campfire in the center of the arena as it will grant a passive regeneration boost which can help you greatly. A Heart Lantern can accompany this if you already have 400 Health.
If you feel confident in the crimson, and don't mind the possibility of other crimson enemies barging in on your fight, you can set up a series of wooden platforms inside the large, circular hollows of the underground crimson. These spaces are usually near 6-10 hearts, giving you a chance to kill 2 or 3 brains in a row using terrain prepared on-the-spot. You can grapple up to wooden platforms, so you can space the platforms up or down more than your normal jumping range if you have a Grappling Hook, already. (Even when holding down, you fall faster with less platforms.)
Alternatively, you can also "box" the arena, by simply putting up a simple wall and ceiling in the big chasm the crimson usually forms, and using materials such as stone or even dirt. You can then set 2-4 wooden platform "floors" which you can then use to maneuver around the arena, and this will prevent other monsters from interfering in your fight. If there are multiple Crimson Heart in the chasm, you can also use this arena repeatedly to fight in.
Keep in mind that the boss will spawn when you smash the third heart, setting you immediately into battle, not when you take your time getting to the arena. Have a path to your arena ready, and try to save any hearts close to your arena as the third heart to smash in a series unless you want to build more arenas every time you want to fight a new Brain. Alternatively, you may try to destroy the hearts from a distance by using explosives.
If you have already made a Bloody Spine, build a little house on the surface crimson, place a bed there and set your spawn, then summon the brain. If you get killed just outside the house you've built, the brain won't despawn, and when you respawn, you can still attack it. This method works great because you don't even need to gear up: all you need is a bed. Note that this may be a bug, so it could be fixed soon. It can also be used to farm the BoC with low tier gear.
Gearing up
It is recommended to have a minimum of 200 Life, which requires 5 Life Crystals; if you've done some spelunking you're bound to have come across this many. You're going to need at least Iron or Lead armor - Lead armor is better as it gives two extra defense than Iron. Gold armor or Platinum armor is a great defensive choice with high defensive values for this point in the game, but considering the rarity of their respective ores it may not be worth it. For those playing during Halloween you can harvest Pumpkins for Pumpkin armor which has the same defence value as Iron armor, but has a 10% damage bonus. (If one is not playing during Halloween you can purchase and plant Pumpkin Seeds from a Dryad, who can easily be acquired by killing the Eye of Cthulhu.)
Weapons
For melee users, the Blade of Grass is extremely good as it deals 28 damage and also inflicts the Poisoned debuff - this is especially good for taking out the Creepers. A Blood Butcherer is a good weapon to have if you've already killed the Brain of Cthulhu due to its large size. The Katana, sold from the Travelling Merchant, is a fast sword with decent knockback and a high critical rate. A spear will be a great weapon for the fight, especially the second stage. Suggested options are the Trident or The Rotted Fork, but a normal Spear is an ok choice. A Flail of some kind is beneficial for taking out the Creepers; the easiest to obtain would be the Ball O' Hurt, but this will require you to make another world with Corruption in it, as it cannot be acquired in a Crimson world. The Chain Knife, a rare drop from Cave Bats, is a good weapon with Flail-like functionality but it cannot pierce. (Hardmode players looking for some quick cash can farm the Brain by using the Death Sickle, which tears through the Creepers and Brain with ease.) Also, The Fiery Greatsword will help, as it deals 36 damage, it also inflicts the brain on fire.
For magic users, you want a minimum of 100 Mana, and 6-8 Lesser Mana Potions. The Water Bolt and Crimson Rod are good choices that are obtained fairly easily which benefit from fighting the Brain in an enclosed area. The Emerald Staff and Diamond Staff are very easy to make and you'll find their piercing ability (especially that of the Diamond Staff) to excel at dealing with the Creepers. A fantastic choice is the Medusa Head, which can potentially kill all of the Creepers in 2-3 shots due to its large range; however, it's very difficult to acquire - it has a 1% drop rate from Medusa, which is a rare and powerful enemy. If possible, you can also get Jungle armor to boost your damage and Mana efficiency. One final choice, is a full set of Meteor armor with the Space Gun; with the full set, the Space Gun will not consume any Mana, allowing you to spray and pray at the brain; the piercing ability of the Space Gun will clear out the Creepers fairly easily. However, this setup requires that a meteor has landed, which is an event that requires you have a Gold Pickaxe at the minimum or explosives to collect the meteor's ore.
For ranged users, Bows with Jester's or Unholy Arrows are good for their high damage. Because they are piercing weapons, both arrow types can hit multiple eyes. However, one can also use Flaming Arrows as a cheaper alternative; the On Fire! effect will be helpful during the second stage. Recommended bows are the Platinum Bow or Gold Bow or Tendon Bow from the crimtane given from Eye of Cthulhu. Guns will be hard to acquire this early in the game, but if you can pick up The Undertaker it's a good weapon to use against the Brain. The Musket Balls that spawn with The Undertaker should be sufficient but you can craft Silver Bullets, which deal a little more damage. If a Meteorite has spawned, you can craft Meteor bullets which will give them ricocheting properties.
Summoner users will have very limited options during this fight; your best option is the Hornet Staff, which is crafted from Bee Wax, a drop from the Queen Bee. Alternatively there is the Slime Staff but this is a 1:10000 chance drop, so it's likely you will not get it. In general, it's advised not to play summoner until later in the game, or unless you are very confident in your skills.
Thrower users can use Shurikens and Throwing Knifes as basic weapons. Poisoned Knifes can be crafted using Vicious Powder and Throwing Knives and deal a little extra damage. Grenades are great choices for dealing with the Creepers but you are advised to watch out for the blast radius, as you will take damage. Molotov Cocktails also work great because of the splash damage and the fire that hits multiple times keep the brain away in its second stage.
Multiplayer
If playing with someone, have one melee player who has The Rotted Fork or another spear to hit the Creepers in the first stage, and have a ranged player off to the side with The Undertaker, a Gold/Platinum bow, or a Gem Staff. In the second stage, the melee player should use The Rotted Fork/Spear to keep the Brain from doing massive damage, and the ranged player should keep shooting it. If you have more than 2 people, have 1 melee and extra ranged players.
Potions
6-8 Lesser Healing Potions will keep you going during the fight. As stated above, you should also get some Lesser Mana Potions if you're a magic-using player. For buffs, the Ironskin Potion, the Swiftness Potion and the Regeneration Potion will do you well. You should try to get the Well Fed Buff in some way; you can get this from Cooked Fish or the Pumpkin Pie.
Accessories
The Cloud in a Bottle will give you a double-jump ability which will greatly help with evasion. Look out for the improved versions, such as the Tsunami in a Bottle, the Blizzard in a Bottle or the Sandstorm in a Bottle.
The Hermes Boots will increase your speed and aid greatly in evasion. You can upgrade them to the Spectre Boots for the added bonus of a few seconds of flight, but this is a long and difficult process to do this early in the game.
The Band of Regeneration is a decent accessory which will slowly regenerate your life; it equates to one heart restored every 40 seconds. Not terribly useful but it's suggested if you have a slot you need to fill.
The Anklet of the Wind gives a basic 10% speed boost, which will help out with evasion. You can replace it with the Aglet which is much easier to find but only offers a 5% speed boost.
The Frog Leg will give you improved jumping abilities, including increased height and automatic jumping but it can only be found by fishing, with an incredibly rare 1:500 chance to get it.
If one is playing in Expert Mode, the Shield of Cthulhu is a great choice for quick evasion, and extra damage in the Brain's second phase.
The Fight
Phase 1
The Brain cannot be attacked in this phase, only the Creepers can. The Brain will teleport to a spot in the arena, then slowly advance towards you. The Creepers will follow shortly after, and will clump up with the brain for a few seconds, before the Brain teleports again and the Creepers will follow. The idea is that the Brain will teleport behind you, which will make the Creepers to fly into you, causing damage. Jump out of the way, either by using your platforms or with an Accessory like the Cloud in a Bottle, then attack the Creepers. Use piercing weapons to speed up the process. When a Creeper dies it can drop either Crimtane Ore or Tissue Sample; make sure to grab them ASAP to stop the drops from despawning or falling down the Crimson's tunnels.
In Expert Mode, the Creepers are much more aggressive. They will actively try to fly into you, even ignoring the Brain if it means an opportunity to land an extra hit or two. They can also inflict a large variety of debuffs; more than any enemy in the game. They can range from being completely unable to use Magic weapons to having your vision darkened.
Phase 2
When all of the Creepers are dead, the Brain will split, revealing a heart with an eye on the inside. In this phase, the Brain will teleport much more frequently, often at a funny angle, and will try to ram into you. He only has 1000 health, the lowest of all the bosses, so he can be killed fairly quickly. Stay calm, and keep using your platforms to jump out of the way. Use a single-targeting weapon to take down the brain, as piercing weapons are less effective against singular targets. If you have a weapon with a high knockback value like a Spear, it will excel at dealing with the Brain, so much so that it's possible to stand in one location and stab it whenever it gets close. When the Brain dies it will drop a large amount of Crimtane Ore, and Cthulhu will be a little less intelligent.
In Expert Mode the Brain will spawn clones of itself around the player at around 1500 health. They serve as decoys, and while they cannot be attacked they can confuse the player as to which one is the real Brain. They start off as barely visible but as the Brain loses more health the clones will become more opaque. At ~300 health the clones will be completely visible. The Brain will also become faster as it loses health, too.
