Six Kings of Pure Color
The Six Kings of Pure Color is a group of level-nine Burst Linkers who each have their own legion and territory. Each king is associated with colors of highest saturation in the "color circle" with distinguished hue, hence the name the Six Kings of Pure Color. Before the defection of the Black King, Black Lotus, it was formally known as the Seven Kings of Pure Color, though Lotus is still known and addressed as a king despite her exile.

Nega Nebulus
 Is known as the Black Legion, under the leadership of Black Lotus. The main goal of the Legion is to defeat the six other Kings and reach level 10 to meet the developer of Brain Burst.

Great Wall
Is also known as the Green Legion. It is led by the Green King, Green Grandee. They control the largest area among the other legions.

Prominence
 Is known as the Red Legion that was originally led by the first Red King, Red Rider. It is currently led by the second Red King, Scarlet Rain. Prominence bore a grudge against Nega Nebulus after the beheading of their first king, Red Rider, by Black Lotus, but after the incident with the fifth Chrome Disaster, they have been on good terms.

Oscillatory Universe
Is a Legion also known as the White Legion. It is led by the White King, White Cosmos.

Auroral Oval
Is a Legion also known as the Purple Legion. It is led by the Purple King, Purple Thorn. They are the most antagonistic legion to Nega Nebulus due to Black Lotus killing Red Rider, the love interest of their leader.

Crypt Cosmic Circus
Is known as the Yellow Legion that is led by the Yellow King, Yellow Radio.

Leonids
Is a Legion also known as the Blue Legion. It is led by the Blue King, Blue Knight. This Legion is known for their direct tactics and close quarter combat in battle.

If you'd like to learn more about the characters, Legions, and more about Brain Burst visit this page: http://accelworld.wikia.com/wiki/Accel_World_Wiki

You can also watch the anime, Accel World, or the movie, Accel World: Infinite Burst! There is also a novel series if you get interested from there.

Disclaimer:
I do not own or claim any of the characters in Accel World.