0.0 are systems outside empire space. They are not protected by CONCORD and there are no in-game penalties for PvP combat. This makes it a very dangerous place. New players are advised to stay out of 0.0 - 0.4 until they are more familiar with the game.
0.0 systems are usually controlled by various player alliances, however, they are sometimes controlled by NPC pirate factions. A player who enters a 0.0 area that an alliance controls could trigger a violent response where the alliance will come and get rid of, what they see as a threat.(See Asakai incident-January 26-27 2013)
They have the rarest minerals, and the most challenging rats, missions and exploration sites. If one wishes to make the most of mining or ratting in 0.0 it is recommended that they join a alliance that owns a particular area consisting of 0.0 systems.