The brands affect the chances of getting certain secondary abilities when you level your gear up. Most brand have one common ability (which is an ability that is five times more likely to appear), and an uncommon ability (which is an ability that has only half the chance to appear).
Amiibo, Cuttlegear, and KOG brands don't affect secondary abilities in any way.
Firefin - 5x chance for Ink Saver (Sub), 1/2 chance for Ink Recovery Sub
Forge - 5x chance for Special Duration Up, 1/2 chance for Ink Saver (Sub)
Inkline - 5x Defense Up, 1/2 Damage Up
Krak-On - 5x Swim Speed Up, 1/2 Defense Up
Rockenberg - 5x Run Speed Up, 1/2 Swim Speed Up
Skalop - 5x Quick Respawn, 1/2 Special Saver
Splash Mob - 5x Ink Saver (Main), 1/2 Run Speed Up
Squidforce - 5x Damage Up, 1/2 Ink Saver Main
Takoroka - 5x Special Charge Up, 1/2 Special Duration Up
Tentatek - 5x Ink Recovery Up, 1/2 Quick Super Jump
Zekko - 5x Special Saver, 1/2 Special Charge Up
Zink - 5x Quick Super Jump, 1/2 Quick Respawn
Amiibo, Cuttlegear, and KOG brands don't affect secondary abilities in any way.
Firefin - 5x chance for Ink Saver (Sub), 1/2 chance for Ink Recovery Sub
Forge - 5x chance for Special Duration Up, 1/2 chance for Ink Saver (Sub)
Inkline - 5x Defense Up, 1/2 Damage Up
Krak-On - 5x Swim Speed Up, 1/2 Defense Up
Rockenberg - 5x Run Speed Up, 1/2 Swim Speed Up
Skalop - 5x Quick Respawn, 1/2 Special Saver
Splash Mob - 5x Ink Saver (Main), 1/2 Run Speed Up
Squidforce - 5x Damage Up, 1/2 Ink Saver Main
Takoroka - 5x Special Charge Up, 1/2 Special Duration Up
Tentatek - 5x Ink Recovery Up, 1/2 Quick Super Jump
Zekko - 5x Special Saver, 1/2 Special Charge Up
Zink - 5x Quick Super Jump, 1/2 Quick Respawn