Percy sees the sarcophagus on the Princess Andromeda. It is described as making the room colder and darker. Luke explains how he is bringing Kronos from Tartarus, and every time a demigod renounces the gods to follow Kronos, a piece of Kronos appears in the sarcophagus.
Kronos reforms in the coffin using Luke Castellan's body with a hole in his chest. This hole heals when Ethan Nakamura pledges his alliance. His palace on Mount Othrys is brought down on him by Nico di Angelo, which stops him from leading the assault on Camp Half-Blood.
In The Sea of Monsters, Kronos' Sarcophagus is described as a 10-foot-long golden casket, much too big for a human. Luke keeps in in his personal cabin on the Princess Andromeda, on a velvet dais. Though the cabin itself was quite beautiful and full of sunlight, the presence of the Sarcophagus made the whole cabin "feel cold."
The Sarcophagus itself is more closely seen in The Battle of the Labyrinth. It is engraved with Ancient Greek cities in flames, and elaborate scenes of grisly death and destruction, pictures of the gods beings trodden under their own chariots, along with temples and famous world landmarks being smashed and burned. The lid is decorated even more intricately than the sides: with scenes of even more horrific carnage and power. In the middle of the lid, there is an inscription carved in letters even older than Ancient Greek, "a language of magic" (possibly the Tongue of the Old Times, which was spoken before the Olympians' birth). It reads: "Kronos, Lord of Time." The Sarcophagus generates an aura of extreme cold, and Percy describes walking next to it being "like walking in a freezer." And when Percy actually touches the lid, his fingertips turn blue, and frost gathers on Riptide.