Pillaging will allow your forces to destroy furniture and tiles in an enemy dungeon. A player's starter dungeon can only ever be raided, not pillaged/conquered, but expansions are up for grabs. Creature have a pillage rating (viewable by hovering over a creature portrait on the creature panel) that show how capable said creatures are at pillaging. Ogres for example are slow, but very good pillagers.
You must launch a pillage attack from an expansion dungeon that can also be pillaged in retaliation. Therefore, it is not possible to pillage from your starter dungeon.