Inner conflict, as I define it, is simply a character's conflict with themself, rather than a character's conflict with the outside world. Inner conflict might be caused by an outside conflict (it usually is), but this is typically because the external conflict reveals something conflicting within the character.
Inner conflict can take many forms. Often, a character holds a certain worldview, but goes through an external conflict that forces them to reevaluate that worldview. Take, for example, the movie Cars. Lightning McQueen, the main character, is a famous race car who is accustomed to living in big cities and being carried everywhere in his trailer. When he gets stranded in the small, rural town of Radiator Springs, he comes to question his own values and morals as he meets the people there.
Or, an inner conflict might take the form of a difficult decision. This would be a major decision, one that calls into question (again) what the character truly values.
What is the purpose of inner conflict? It helps us connect with the characters more. Connecting with characters is critical to good storytelling; we don't care what happens to characters we don't care about.
Inner conflict makes your characters more human. Real people have inner conflicts. In fact, most of the inner conflicts in literature are real conflicts that many real people face.
Having a character go through such an inner conflict allows us to connect with them. Have you ever battled an evil wizard? Have you ever blown up a Death Star? Most of us haven't . But we can relate to the situations these characters are in, largely because we can relate to their inner conflict. We can think about a character's dilemma: Would I hand myself over to the guy who wants to kill me in order to save my friends? and ask ourselves if we would do the same. We can watch Luke's inner turmoil as he is tempted by the power of the Dark Side and compare it to our own temptation.
TL;DR Inner conflict is simply a conflict that occurs within a character's mind, usually involving their morals/beliefs/values. All types of conflict are engaging if they are written well, but inner conflict is one of the most relatable, because we all experience it.