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.