Each civilization is really unique at doing something. The super abilities are:
Super Diplomat. Civilization is able to secure much better deals through negotiation that other civilizations.
Super Dominator. When they go to war, they receive a group of Corvette class ships. The number they receive is based on their military power.
Super Organizer. When they are attacked, all races of the same ethical alignment will come to their aid.
Super Warrior. When in battles, they get the first strike ability. That is, their ships attack first and if their opponent survives only then do they return fire.
Super Breeder. If their approval is 100% their population increases at 4X the normal rate
Super Isolationist. Can colonize barren worlds and no foreign ship can travel more than 3 spaces per turn in their area of influence.
Super Trader. Gets all the trade techs at start of game and gets 2X from trade.
Super Manipulator. Is very good at getting other races to go to war with each other.
Super Hive. Unlimited range and all factories are built in 1 week.
Super Adapter. Starts out with aquatic and toxic world technologies.
Super Annihilator. Can build spore ship which when used on a defenseless colony will eliminate all population and turn it into a toxic world of the same class.
Super Spy. Gets an agent to start with and worlds are immune from agents.