What is a Number Theorist
Solution 1:
I think non-mathematicians are often more interested in the "why" and "how" than the "what", so I'm usually very light on mathematical details (unless they ask for it). I try to talk more about the motivations, history and fun facts of it, developping more or less detail depending the listener's (apparent) interest. My usual answer is along those lines :
In a nutshell, Number theory is the study of the integers. Although their definition is quite straighforward, there are still many unanwsered questions about them. If they ask for an example, I usually talk a bit about primes numbers : I explain we know stuff (that there are infinitely many of them, we can even roughly count them) but it's difficult to actually find them. Sometimes, I also mention that some seemingly simple problems like Fermat's last theorem took centuries to be solved.
One reason for this difficulty might be that there's a bigger picture that we still don't see. So being a number theorist is trying to get some perspective. Very often, this requires taking a detour, like creating and mastering sophisticated new objects which retrospectively sheds some new light on old ones. If they ask for an example, I talk about how some new numbers (negative, complex...) were created to solve equations, and mention that number theorists have invented a whole lot of other "exotic" numbers for other purposes (finite fields, algebraic integers, quaternions, $p$-adic numbers...).
Now number theory is interesting because integers are at the heart of mathematics, so understanding them might lead to advances in mathematics as a whole (which might lead to advances in science ?). Another reason is that we use it in cryptography because it provides problems that are difficult to solve. And on a more personnal note, my motivation is also that I find it beautiful and quite fascinating.
Solution 2:
When I explain what I do to a nonmathematician, I usually briefly explain what a prime is (if they don't already know) and talk about how there are infinitely many of them.
If it turns out that they asked the question out of something besides cocktail-politeness, then I might continue to say that this is sort of like starting with the number $1$, adding $1$ a whole lot to get the sequence $1, 2, 3, \dots$ and seeing if that sequence has infinitely many primes. But we may consider other sequences too - it turns out that other sequences, like $2,(2 + 3),(2 + 3 + 3),\dots = 2, 5, 8, \dots $ have infinitely many primes too, while sequences like $2, 6, 10, \dots$ don't.
Let's be honest - usually people have tuned out by now. But let's suppose I'm talking to my girlfriend or something, and thus that she would feel awkward if she cut me off now (not saying this has happened or anything ;p). So I might continue to say that it turns out that one of the ways to understand primes in sequences is to understand what's called the Riemann Zeta function $\sum \frac{1}{n^s}$. This is sort of cool, and just a hint of the connections with certain special functions and the behavior of numbers.
Now I cheat a little, and mention something that people feel is a bit more approachable. The zeta function appears in other places too, like Zipf's law. And Zipf's law applies to many things besides languages - it tends to closely describe things like the major players mentioned in newspaper headlines. Some social sciences even claim it for their own (here, it is used to say that the size distribution of cities must fit a power law).
The binding idea here is that arithmetic functions, or other functions for that matter, sometimes have things to say about topics which they might not seem to describe. And as a number theorist, I look into some of these families of functions and how they behave.