Super Saiyan 6 is the sixth form of a Saiyan Warrior, achieving power level even beyond that of a Super Saiyan 5, this form is taken by those who can at least go Super Saiyan God, upon unleashing rage (similar to a Super Saiyan), the Saiyan's true powers will be unleashed, thus transforming into this form, giving a whole new unimaginable amount of Power Level, that can easily outmatch the God of Destruction, Beerus.
The appearance of the Super Saiyan 6 doesn't change much from a Super Saiyan 4, where instead of Red Fur, the whole upper area is covered with Golden Fur, and the hair is turned into golden like that of a Super Saiyan 3, and shaded red area under the eyes similar to Super Saiyan 4.
Whoever uses the Super Saiyan 6 transformation, he will be a tough opponent, as the power levels of this form is around that of a full powered Super Saiyan God. Considering that Super Saiyan God Goku wasn't able to beat the God of Destruction, Beerus, the Super Saiyan 6 transformation still would not be able to beat a God of Destruction, but would at least be a match. Surpassing even a Super Saiyan 5, the Super Saiyan 6, is assured to be one of the toughest and strongest transformations in all Dragon Ball Z.
Super Saiyan 6 eventually changes the personality of whoever uses it, into a cocky-over confident character, even Goku will turn that way. But when serious, the character changes into a wild personality, stating that the Super Saiyan 6 cannot be defeated under any circumstances.
Super Saiyan 6's creation was after the Battle of the Gods movie, considering that Goku will make a comeback to Beerus with an extremely powerful transformation that can brush Beerus aside like an insect and forcing his attendant, Whis (who is much more powerful than Beerus) to a fight, eventually ending up in a tie. It was revealed then that Goku (who went Super Saiyan 6) was only using 50% of his Power.
Vegeta, enraged that he felt he will never be able to surpass Goku, even after he admitted that he can't during the Buu Saga, he refused to be surpassed by such a huge gap, training in the Hyperbolic Time Chamber for over 3 Years, Vegeta asked Goku for a fight. Goku, going Super Saiyan God, was interrupted by Vegeta, and was asked to go Super Saiyan 6... Goku responded and went Super Saiyan 6 according to Vegeta's Wishes, then Vegeta, shockingly went to a form surpassing a Super Saiyan 6... Goku then, was happy that he will finally find someone he can enjoy fighting with. After being punished by the Powered-Up Vegeta, Goku as well went to the same form Vegeta did. Shocking Vegeta, then both going 100% Strength. The fight was thought to end in a tie, but Goku was able to make a gigantic Kamehameha, it didn't kill Vegeta, but forced him to give up.