A thesis is expected to demonstrate both a thorough understanding of a specialized topic and the ability to explain it in a clear and well-organized way. Theses sometimes do contain publishable original research, but it is difficult to get that kind of result in the time available. Quite a few students enter Princeton with advanced placement in mathematics, but it is not at all necessary for majoring in the subject. In mathematics it is possible to get very good grades in high school and fairly good grades in underclass courses on the strength of one's ability to apply rules in a somewhat mechanical way. Actually, it often makes sense to learn a procedure and practice its application before completely understanding the theory behind itotherwise it may be hard to see the point of the theory.
People study mathematics for various reasons.
Some do so simply because they find it interesting and enjoyable for its own sake. However, training in mathematics is good preparation for a number of careers. As with any subject, there is the possibility of an academic career. Knowledgeable teachers of mathematics and science are badly needed at all levels. Graduate schools of business and departments of computer science and statistics generally look favorably on applications from mathematics graduates, especially those who also have some courses in the relevant specialties.