As you play through Story Mode, you'll occasionally play what are essentially button matching quick-time-events. Win or lose, the story continues anyway.
They don't have to be done perfectly to win, but if you happen to lose, hit , quit to menu, go back to story mode and continue to try again. If you've already completed the game, you can use chapter select to replay any of the minigames. Wait until the game auto-saves (icon in bottom corner) before quitting.
If you didn't beat all of them during the story mode you can go back to each one individually without having to do the story all over again. The hardest one for me was Green Arrow vs Grundy until I realized that you can beat it by letting the cleavers and chairs hit you and just focusing on hitting grundy with your arrows. Below is a link to what chapters the minigames are in.