DISCLAIMER: This page is still being heavily translated from the Japanese wiki, any incorrections should be changed in the near future.
Star Cores are the items that determine what dungeon you will create. Click on the Star Core for an in depth guide on what they do. They can be found all around the world, as well as bought from a magic store vendor in most towns. They consist of three types:
Red Star Cores determine the type of monsters you will face in the dungeon, as well as the boss itself. Pay close attention to this one most of all, as choosing the wrong one can leave your party unprepared for what may lie inside. Certain bosses carry certain loot, so be sure to choose your Star Core carefully.
Yellow Star Cores determine the type of environment you will be fighting in, and the map layout. Although most dungeons are typically similar in layout and pattern, this Star Core will determine boss location, key placement, doors/exit/entrance location as well. It is also assumed that this Star Core will also determine the existence of an elite/boss, or not.
Blue Star Cores determine the type of loot you will receive from ordinary monsters in the dungeon, as well as from the chests. Also, certain Blue Star Cores are heavily weighted towards abundant chests, others focus more on more loot from the monsters themselves. If you're trying to find a specific item, then choose carefully. Each one has it's own loot table.