A Grave is a placeholder object meant to show a dead unit.
When a battle takes place in an overworld room (like Axel's Home or Holt Village) and a NPC is killed in that battle (for example the fight with Friday MKII) they are removed from the "room" and replaced by a grave with their name on it. Characters that are important to the storyline are not present during the battle so they can't be killed. Shopkeepers, including the Item Worlder or Dark Assembly NPC, can be killed, resulting in their facilities becoming unusable.
If by some accident you kill one of the NPC's, a quick save and reload will fix the problem. The save and reload was mentioned in the official strategy guides for the game.
In Disgaea 3 and Disgaea 4, destroying a Grave in battle will spawn a Ghost enemy at its location.