So you've been in a long and tedious battle with That One Boss, and it finally looks like things are going your way and that you have the upper hand. You breathe a sigh of relief: you think you've gotten the boss's pattern down pat, you've learned how to effectively deal damage to it and dodge all of its attacks at the same time. A wave of relief washes over you as you finally get its health meter down to half -- now, you just have to do it all over again once more, and you're the winner. You got this, man! You got this by the ass!
But wait! Suddenly, when that health meter hits the halfway point, something happens. Oh, crap. Now the boss is pissed. A quick cutscene shows his body changing to a brighter color. Steam shoots out of his head like a smokestack. His body armor breaks off to reveal his creepy inner organs. He roars at you with a newfound rage you didn't know he had. He stomps on the ground and pieces of the floor fall away. He unleashes brand new, devastating attacks. His movement speed triples, and he fills the room with ten times the projectiles you're used to. Good luck, the boss you're fighting has just Turned Red.
Also known as "pissy boss mode", turning red is a form of Difficulty Ramp applied to enemies and bosses alike. It's kind of a second wind that enemies get when they've been damaged too much, or when they're the last enemies on the screen.