I happen to be a psych prof, so maybe I can help.
1. You don’t need permission to go to office hours. Although a heads-up helps, you can just show up. Some students will make appointments during this time to guarantee a spot, since office hours can fill up quickly with drop-ins.
2. Great suggestion above about talking with other students. Learning to use all sorts of resources to solve problems is a college goal beyond just the academics.
3. What do the notes you’ve taken say about theories of personality? What does your textbook say? Your school librarian can assist with finding quality online resources if you’re unfamiliar with them.
4. It sounds like your prof wants you to choose a theory of personality (there are many) and then apply it in a concrete way. To do this you’ll first need to understand a single theory in-depth, and only then can you apply it to a real-world situation.
5. The prof didn’t give you zero instructions. It seems like you’re having trouble understanding his instructions, but they are there. You listed them in your post. Sometimes instructions may feel poor if you don’t yet understand the basics.

Ex: Trait theory. Categorizes attitudes and behaviors into “traits”. Combinations of traits give rise to a better understanding of one’s personality. Potential application: Someone with a, b, c, etc. traits might have such and such a personality. This personality type could describe why the person has behaved/feels in x way in x situation. It could predict how the person would behave/feel in x future situations.

It’s also worthwhile to discuss the limitations of your chosen theory. It shows your ability to think critically and analyze information, rather than just regurgitate it.

—Meow