A strong character analysis will:
- identify the type of character it is dealing with.
- describe the character.
- discuss the conflict in the story, particularly in regards to the character’s place in it.
Often the characters are described in relation to the conflict within the story.
Conflict can be many things:
External –
man vs. man: This is the protagonist versus the antagonist. Snow White versus the Wicked Queen.- man vs. machine: This is when the machine is the enemy. Many robot-centric novels have this issue. (This is sometimes considered a subset of man vs. man.)
- man vs. nature: Robinson Crusoe on the island. Hansel and Gretel lost in the forest.
- man vs. animal: Captain Ahab versus the white whale in Moby Dick. The wolf in “The Three Little Pigs.” –Usually the animal is a predator and the man has become prey for some reason. It could be humorous, though, the man is trying to catch the dog, who runs away and has the main character chasing him all over creation. (This is sometimes considered a subset of man vs. nature.)

- man vs. fate or destiny: Sleeping Beauty can’t help pricking her finger. A man who has been late several times (due to circumstances beyond his control) gets in a traffic jam and is an hour late to work and gets fired. The fact that it has happened several times and is not his fault is the crucial point.
- man vs. society: This is when a character battles societal norms. Winston Smith in 1984. Huck in The Adventures of Huckleberrry Finn.
Internal –
- man vs. himself: This is when the character has an ethical dilemma, stealing to feed his family or watch them starve. Lie to the government and save the people in the basement or tell the truth and have them taken away. This is the cartoon equivalent of the devil and the angel on either shoulder.

- man vs. his mind: This is the character with internal problems that are not ethical, but mental. An example, as was pointed out in the comments, is the character with schizophrenia or one who is bipolar. How does the character deal with his/her limitations? What do they have to overcome? How do they overcome it? Is it harder or easier to overcome something that is a part of the character than it is to overcome something that is outside of the character? Thanks to commenter Kenneth for the additional aspect I had left out.
Other sources on this and related topics:
How to Write a Character Analysis: Introduction
How to Write a Character Analysis: Body Paragraphs
How to Write a Character Analysis: Titles
How to Write a Character Analysis: Conclusion
Questions for Literary Analysis: Theme
Questions for Literary Analysis: Setting
Questions for Literary Analysis: Point of View

