The Eye of Cthulhu is probably the first boss a new player will fight in a new world. It is the "easiest" boss even though the EoW is easier. It is a very non-essential boss and the only reason one would kill it is to keep it from surprising you later in-game.
Weaknesses: Has the lowest health of all the pre 1.1 summonable bosses.(not counting King Slime of course as it has 2000 health)
Strengths:Spits out minions in the first phase, has 2 phases, has a different skill set in each one, does high damage.
How to Kill:Assuming you have at least an Iron Bow and a Musket(250 ammo) this is your guide. If you have access to late-game equipment then this boss would go down in seconds.
Build the same arena for Eater of Worlds and you will need a bow and a gun.
It is important to keep your distance from this boss as it could destroy your strategy if you allow it to get close, especially in the 2nd phase. Just keep shooting your Musket or Bow(flaming arrows) at the pupil in the first phase as thats where the Servants of Cthulhu will spawn from. In the second phase it doesn't matter where you hit.