Dungeons are the cave-like places in which you crawl from room to room by tapping on the desired room that you want to go to. These Dungeons can consist of an X amount of floors and can have various levels of difficulty and monsters depending on the one you go to.
A Spirit Totem can also be obtained (usually on the final floor) of a dungeon. Each dungeon will more than likely have one.
Some breeders will be lurking in these Dungeons, as well. Breeder fights will have red outlining the monster to help you clarify that that fight is indeed a breeder fight and not a normal monster fight.