To unlock Cognito Mode finish all the puzzles in Puzzle mode. It is okay to look at hints constantly to solve all the puzzles. Cognito Mode is basically Puzzle Mode with no hints.
Finity mode
Complete 280 levels in Endless mode.
Hyper mode
Complete 8 levels in Action mode.
Original mode
Select the "Play Game" option. Then, select the following options in order repeatedly: "Classic", "Action", "Endless" then "Puzzle". After a several cycles, the message "Please Wait" will appear. A version of the game similar to the original Bejeweled will begin.
Remove backround grid or change jewel types
Hold LB, LT, RB, RT then press B button to remove grid or press A to change jewel types.
Tasks list
Get to level 9 in action mode - Action mode
Get a x8 cascade during any game mode - Cascade 8
Get to level 18 in classic mode - Classic mode
Collect 50,000 in cognito mode - Cognito mode
Finish level 280 in endless mode - Endless mode
Get to level 9 in finity mode - Finity mode
Get to level 9 in hyper mode - Hyper mode
Make 1,000 power gems across all your games - Hyper1000
Twilight, Hyper, Cognito, and Finity mode - Multi mode
Make 10,000 power gems across all your games - Power10000
Finish all 80 levels in puzzle mode - Puzzle mode
Finish level 10 in twilight mode - Twilight mode