I work part time as a tutor.
This is a combination of the words 家庭 and 教師. A "teacher" who only teaches when you're with your "family" is known as a tutor. They don't teach you at school, they're a private teacher that comes to your home to help you learn.
This is a jukugo word that uses the on'yomi readings of the kanji. You should be able to read this on your own.