Depending on the type of tutoring and the needs of the student what a tutor does can vary. I will give you some examples, which I hope will be helpful to you.
If a student is having trouble with a class, say an algebra class, then the tutor will focus all his/her energy on improving the performance in the class. This may mean helping the student with homework, reteaching and re-explaining concepts that were covered in class, providing extra practice, and demonstrating to the student how to study for tests. Generally this type of tutoring is done while following the curriculum of the school.
Some tutoring programs offer their own curriculum, somewhat like an additional subject or class. The theory behind this is that what the student learns in the tutoring curriculum will transfer to the school classes and make them easier. These types of tutoring programs train their tutors in the specific system.
If a student is on summer break or needs enrichment, or is studying a class outside of school, then the tutor will need to come up with materials to use. The student's teacher and parent can help select the materials, or the tutor may prefer a certain set of materials for the purpose.
Most of all, what a tutor needs to be able to do is connect with students. You need good communication skills and a knack for getting along with young people. Teaching and tutoring experience is essential too, because as anyone who has tried can tell you, teaching/tutoring a subject is different than doing well in it as a student.