The tutor is who you hire. The benefits of a tutor are that you have a higher chance of winning a battle (depending on their stats), you get 'tutor bonus', which is extra money when you battle, and when it gets hired by someone else, you earn your money back plus a little extra. Also, when they buy new dorms, you get a little money.
Your pupil is the person who hires you. You can talk to him/her (and your tutors) through Tutor Chat. Every time you get hired, you get money, limited at $1 million per hire.