Across the story mode you will need to complete a minigames where you need to press the buttons prompted in the screen and win them, If you fail some minigame you can retry on Chapter Select.
There is no need to finish the follow fight nor to complete the following scene, you can quit immediately after the minigame ends.
Here is the list of mini games:
Chapter 1: Batman vs Lex Luthor
Lex Luthor will drop down on one knee once you've defeated him
Chapter 2: Green Lantern vs Sinestro
Green Lantern will hit Sinestro with a bus
Chapter 4: Joker vs Nightwing
Joker will gain a full hand of cards and throw them all at Nightwing at once
Chapter 5: Green Arrow vs Solomon Grundy
Three rounds of shooting Solomon Grundy with arrows
Chapter 8: Batman vs Green Arrow
Completed once you've blocked Green Arrow's triple shot
Chapter 11: Wonder Woman vs. Bane
Completed after the button mashing to block Bane's bullets
Chapter 12: Superman vs. Black Adam
Simply watch the bar above the 3 Superman symbols