Some of the students I have tutored in math just want to get through one last course and, so they believe, get it over with for the rest of their lives. Others enjoy it and plan on further study. Some of the latter have very specific career paths in mind, most commonly engineering.
To those who are less clear on how they might apply mathematics, I offer this Science Daily
book review called Mathematicians Reveal Secrets of the Ancient and Universal Art of Symmetry. Simply reading the article offers some insights into the relationship that mathematicians have with both describing and understanding patterns that have fascinated people for centuries. For those not inclined toward engineering, the book sounds like it could give a young person a good example of the breadth of subjects to which mathematics can be applied.