It starts when Father Flynn says that the reason Donald had alcohol on his breath wasn't because he gave it to the boy, but because he caught the boy drinking the alter wine. He didn't want to report the incident because he thought that would look bad on the perish, and the cardinals (or whoever is the next highest level, I'm not sure) would look down upon Sister Aloysius because she is the principle of the school.
Later on, when Father Flynn is having a conversation with Sister James in the courtyard, he attempts to justify his actions, claiming that there were "circumstances beyond your knowledge." He also seems to have interest in only one boy (Donald), as seen when he asks Sister James how he is doing in her class. When she turns the conversation to Sister Aloysius showing concern because she cares about the child, Father Flynn replies with frustration, "Its me that cares about that boy." Even at the end of the scene, there is a large black crow (sometimes seen as a symbol of one's conscious), which he yells at to be quiet. This could be that either his conscious is telling him to go to Sister Aloysius and explain what happened with the alter wine (thereby clearing his name if he is, in fact, innocent), or it could be telling him that he knows what he did was wrong, and he should simply come clean with it and take the consequences.
Later, Sister Aloysius calls Father Flynn into her office. When she can't coerce a confession out of him (using evidence such as the alcohol on Donald's breath and the fact that Father Flynn had been through five perishes in the last three years), he turns offensive and starts to verbally attack her, rather than turning defensive, as someone who is hurt from the accusations would have. When she says that she is going to call someone higher up, he stops her.
Finally, when Sister Aloysius calls Father Flynn into her office for the last time, she lies by telling him that she called his previous perish. She tells him if he doesn't want to lose everything, he will ask for a transfer, which he later does. If he hadn't been guilty of something, he would have continued to fight her, but I doubt (no pun intended) that he would have transfered for the sole reason that he was tired of putting up with her accusations, as some others (not necessarily in the class) seem to believe.