A Cauldron is a cave/dungeon that can be explored. Cauldrons get marked on the map when you've ventured close to one or you have overridden the nearby Tallneck. These are marked on your map with what looks like the head of an arrow with a symbol in it (looks like ^). Each cauldron will contain a different symbol within the ^.
There are five cauldrons scattered throughout the map. Once you approach a Cauldron you'll need to open the door or find the hidden entrance. There are usually a lot of enemies in these Cauldrons and you'll need to fight your way past them to reach the core, which will usually consist of a boss battle. After defeating the enemies at the core, the core itself will rise out of the ground and you'll need to walk up to it and press Triangle to override it. After accessing the core's information, it will give you a new set of machines to override. Complete each cauldron to unlock the trophy.