The Brain of Cthulhu is a CrimsonBoss that resembles a large flying brain. It spawns when players destroy 3 Crimson Hearts, or when a player uses a Bloody Spine.
20 to 30 Creepers spawn along with the Brain and orbit at high speed. All Creepers must be destroyed first: As long as even one survives, the Brain cannot be damaged. Once the Creepers are destroyed, the Brain itself becomes vulnerable to attacks. The Brain cannot spawn new Creepers.
The Brain of Cthulhu is the Crimson counterpart to the Corruption's Eater of Worlds. Much like its cousin, fighting it is essential for progression, as the Brain and its attendant Creepers are the only source of Tissue Samples and usable quantities of Crimtane Ore, which are required to craft advanced tools and armor.
The Brain itself is initially invincible. It teleports around the player, drawing nearer before teleporting away again in a random direction while its Creeperminions attack. The Creepers do not teleport along with the Brain, but are instead pulled between the player and the Brain in large, sweeping orbits. Once all Creepers are defeated, the Brain enters its second phase.
Second phase
In the second phase, the Brain becomes vulnerable to attacks, and aggressively attempts to ram the player before teleporting away.
As Creepers drop Tissue Samples and Crimtane Ore fairly frequently, it's technically not necessary to defeat the Brain of Cthulhu to gain access to Crimson-tier equipment, only to kill a couple of Creepers and secure their drops.
Creepers have low individual health and are especially susceptible to the On Fire! and Frostburndebuffs, so weapons that inflict those can make quick work of them.
The Brain of Cthulhu is the only boss susceptible to knockback. Weapons with high knockback can keep both Creepers and the Brain from closing distance between teleports.
Unlike the Eater of Worlds, the Brain of Cthulhu doesn't run away and despawn if you leave the Crimson, therefore you are able to summon it and lead it to a nearby arena in a less hilly biome.
Trivia
The Brain of Cthulhu, along with the Eye of Cthulhu, is based on the character Cthulhu from the works of the author H.P. Lovecraft.
The Brain of Cthulhu has the least HP out of all the bosses, with even less than the Eye of Cthulhu in its second form. However, if counting the health of its creepers, it has more health than King Slime.