TITLE: What is a Number Theorist
QUESTION [11 upvotes]: I often find myself facing this problem when asked by non-mathematicians about what I do. When the answer, "I'm a mathematician," doesn't suffice and I have to reply that I am a number theorist, I do not know how to explain to a nonspecialist what number theory is. The answer, "I study numbers," feels rather unsatisfactory to me at times like these. My thinking is that I'm probably not the only one who has encountered this. So my question is, how does one explain what number theory is to a non-mathematician?
For the purposes of this question I think it is reasonable to assume our audience possess a standard high school education.
EDIT: Some of the answers seem to go off tangent so I guess I should emphasize that the question really is, "how does one explain what number theory is to a non-mathematician?" and not how should a mathematician socialize with a group of non-mathematicians.

REPLY [7 votes]: Consider what people are looking for when they ask you this. People are looking to know about you, not for a lesson on prime numbers.
Suppose I meet folks in non-geeky jobs like finance or law or real estate. When I ask them what they do for a living, and even if I really want them to go into it, I'm not looking for financial methodologies or legal precedents or the way a home's value decreases over time while it's on the market. If you are trying to get to know a person (primarily) or what they do (secondarily), those things are tertiary.
When I am asking what someone does, I want to hear something like "My team is responsible for capital budgeting at GBX." They may tell me how excited they are about finance, or how they're studying for an MBA part-time, through GBX's tuition-reimbursement program. Or maybe they'd tell me it's just a job, and they're think of going elsewhere. People who are really good at this will have a few stories ready to share, and they're often not even directly related to their field. It may be a story about a coworker, or a client, etc., to engage your interlocutor for a few seconds. 
So instead of explaining number theory, you're better off saying you're a professor or a consultant or whatever it is you do, and telling a fun anecdote. Mention that you're busy grading papers all the time, or tell something funny a study wrote on a test. The point is to tell people about yourself.
As @Auke and others pointed out, a lot of people don't understand what Mathematicians do. I don't mean that they don't understand what mathematicians study (which is probably true as well), but moreso, what their jobs looks like. When pressed on that subject, consider again avoiding the specifics of your area of research and speak to your tasks or assignments. For example, do you spend a lot of time teaching? Doing research? What does doing research look like? How do you collaborate? Or if you are working in industry, what is your role there--writing software? Doing some sort of analysis?