A conflict which a character faces which puts them at odds with their inner self is called internal conflict. Internal conflict is where a character fights with their own morality. Many times, a character must decide what is the best thing for themselves, others, and the world in general. Unfortunately, issues such as these are not always easy.(An example of internal conflict, generically, would be a person on a diet and desiring chocolate cake. The person wants the cake, but it would not be good for their diet. Here, the person must decide what is more important: adherence to the rules they have set for the diet or their desire for the cake.)
There are multiple examples throughout literature which speaks to the internal conflicts which characters face. Here are two textual examples:
1. Speak-- Melinda is struggling with her own inner demons. She recognizes her depression, but finds it difficult to speak with others about it. Knowing that she needs help does not outweigh her depression. Therefore, Melinda is in a battle with her inner self.
2. Macbeth--Macbeth fights,internally, with his ambition and his sense of right and wrong. He wants to be king, but he knows that murdering Duncan is morally wrong.
As for more general examples, internal conflicts can be seen when a character holds a secret which they know, if told, would save another character. Another example would be a character who is caught between two different loves--they are drawn to both people, but do not know how to choose.
Basically, an internal conflict is anything a character struggles with in their mind or heart.