The tournament is the little trophy icon (the one from the achievement tile) on the left side of the level selection screen. It is active for 72 Hours beginning on Friday and going through the weekend. You don't need to do well or even get that anywhere in the level. As soon as you start playing, the achievement should pop and then you could die or play as far as you can.
The tournament is a looping stage where the difficulty of the loop continues to ramp up until you are destroyed.
EDIT: According to OhMyGoth1, you need to complete missions 1 through 3 to unlock Tournament Mode.