For me, this is possibly the most exciting part of the whole writing process, even more so than the story itself, because the characters make the story!
Actually, I create my characters before I begin coming up with a plot. I'll have a general idea of the sort of world and story I want to create the characters for, but the plot comes through them and their desires. Perhaps that is a bit different than how most writers do it, but I wouldn't have it any other way. We are all familiar with the quote by Erin Canavan Marks, "You are the hero of your own life's story. You just have to learn to tell it that way." Well, Ms. Marks has the right idea. The story begins with the person, the character, not the other way around. This is why character creation and development is so important and exciting.
So, first think of your main character, the one the story will focus on most of all. For me, that is usually a strong female character, because, well, I'm a domineering female character. Write what you know and all that haha. I usually ask myself these questions first:
1. How old is she?
2. Does she have a family? Why/why not? What are they like?
3. Is she an extrovert or introvert?
4. What class is she (role playing thought process meaning are they a rogue, fighter, barbarian, wizard, etc...)? What skills does she have as a result of this class?
5. What is something quirky about her (e.g. she is highly competitive, or will drop everything to pet a dog...maybe even believe in some crazy superstitions)?
6. What does she want out of life as a whole? For example, Eithne (one of the main characters in my book) only wanted to be allowed the freedom to learn how to fight and live her life as she pleased. That's it.
7. What is the cultural background of your character and how has it affected her? Rebel? Conformist? Indifferent?
After I have answered these questions, I usually have a good grasp on who she is and what she is about. My next step is to create the other characters in their life. This means family, friends, enemies, frienemies, etc... You know, the usual. Since I am an animal lover as well, you will notice that all of my stories will at the very least mention a specific animal and it's relationship to the main character(s). I also frequently base characters on people from my actual life. For example, my parents and my brother were both heavy influences for characters in the book I am currently writing. The people in your real life most likely already have some pretty amazing traits, use that inspiration.
Taking queues from your own personal interests/desires/fears can give your characters a whole new level of depth. Don't be afraid to express yourself through your character. Everything in a story you have written is a piece of you after all. Be yourself. Writing is a lovely opportunity to express yourself and live out your dreams through your characters. Enjoy it.