I have been a math tutor for elementary through middle school. I've also had the opportunity to talk to tutors and teachers, some more successful than others.
~Empathy. If you have been there, if you have struggled with that class, if you can understand what exactly about a particular problem / concept is tough, tell them. Despite the fact that on the outside, tutoring seems like an easy transfer of knowledge from one person to the other, it's also really emotional. Students are scared about their grades and their futures. they want someone to sit next to them and say, "I know how you feel, you can do this, you are going to be successful." Especially in math, where students come into a session assuming that because they "have always sucked at math," they will never get it. In many cases, math students struggle from these kinds of internal mental blocks more than the actual level of difficulty. I have many new angles of approaching a problem. Students come to tutoring because they don't understand their teacher's examples; give them as many different examples and explanations as it takes for something to click. In this process, your knowledge of the subject will get much deeper as well.