The Council is a group of many alternate reality versions of Reed Richards, banded together to do what every Reed wishes to do: Solve Everything. The alternate reality versions have a wide array of powers, technology, power items (such as Infinity Gauntlets and Ultimate Nullifiers) and some have no powers at all. They are all extremely intelligent.
Together, they do great things, such as feed the starving and stop threats to all realities. Their darker side includes the Vault, an area where they keep the various Doctor Doom's from other realities, which they lobotomize. Reed Richards of the 616 continuity was welcomed to join in their efforts to make the multiverse a better place.
When the Celestials attacked, it was 616 Reed that saved the day, bringing a weapon that instantly killed one of their attackers and eventually chased them off...into other realities where they could rebuild their forces and recruit Celestials from other realities. He soon learned however that the price of solving everything is everything, and most of the others had lost ties to all their family and friends. Reed decided that he would rather stay with his family and never returned.
Unfortunately, they would soon return to haunt Earth-616 thanks to some assistance from Valeria, who found the remaining members of the Council hiding deep inside their Citadel from Arishem the Judge and three of his subordinates. Though Valeria attempted to get all of the Councilmen out, only four made it to Earth-616 alive.
They soon went into hiding and began plotting the destruction of The Celestials using a reality-destroying weapon called "Sol's Anvil" which they had once used to destroy an alternate version of The Beyonder. They have since begun interacting with the Universal Inhumans, Annihilus, Mole Man and the Moloids and the Uhari to bring about the War of the Four Cities in order to gather together the components required to build the Anvil.
While most of the Reeds succeeded in getting what they wanted, one of them was killed by The Universal Inhumans, who believe they can "match" the Council's plans, something the other Reeds seem to be betting on. Shortly afterward; The Invisible Woman, Spider-Man and Alex Power tried stopping a Chordai/Mala allied attack on Old Atlantis only to find themselves under siege from Mole Man, his Moloids and one of the Reeds as well.