They were added in the Gamecube remake.
Basically, when you kill a zombie, his body will persist, and as time passes in the game, the body will mutate and turn into a much stronger, fiercer and faster zombie. You can tell if the body has transformed if the skin has turned red and his nails have grown out into claws. You can also see him move very slightly if you stand still and look closely.
As you move close, he will soon spring up and attack you, like he's lying there and waiting to ambush you.
To prevent Crimson Heads from happening, you have to either have destroyed the head of the zombie when you killed him, or you need to douse the body in kerosene and set him on fire with a lighter. The thing is though that the kerosene flask takes up an inventory space, and can only hold two doses of kerosene at a time, further there's only a couple of kerosene cans around the mansion, and they all have limited content. Further, if you're too late and attempting to burn a body, the Crimson Head might spring up on you as you're pouring the kerosene, wasting your dose.
One body in a hallway is scripted to already be transforming early in the game, and one encounter is set up later with an extra strong one, as a small bossfight. All the other zombies and bodies come down to chance.
It is, I think, one of the neater additions to the remake, and it really changes up the gameplay, as you need to weigh the use of 1 inventory space as Chris (who has a lighter in his 'special' slot), and 2 inventory spaces for Jill (where the lighter takes up inventory space). You either make it a point to systematically clear places out and burn bodies as necessary, going back and forth to refill your flask, occasionally lucking out and destroying zombies' heads, saving you doses, or you don't, and risk numerous dangerous encounters at numerous bad times, at the advantage of a less crowded inventory.
I like both the tactical element and the horror element that this all adds.