Their are two things I say to people when I am in such a situation:
- I think of math as being divided into 3 parts: Algebra, Analysis and Topology. Each of these comes from starting with a set and looking at different types of structure on it. Algebra is about taking two things in the set and asking how you can make another element of the set given those two things. (or maybe take 3 things! etc) Analysis comes from taking two things in your set and asking what the distance is between them. Topology is about seeing when two things in your set are close two each other (it gets hard to convince them this is different from analysis, but it is doable).
Now first I should mention that this is a very simplistic description and probably incorrect, but you only have ten minutes and the audience can feel like they learned something about the field, or at least how it is put together, or was at least 100 or more years ago. This also might not appeal to you since I haven't said anything about set theory, but this does give you a jumping off point for talking about what you can do when you don't have any of those more "sophisticated" structures lying around.
- I study algebraic topology, so I try and talk about how you might try and differentiate two geometric objects or rather determine if they are in the same homeomorphism class. First you have to say when you will be thinking of two things as the same. This seems a bit strange to people, but remind them of congruent triangles and how natural it is to think of congruent triangle as the same thing when they are in different places. Then tell them it is a lot like that, but with different rules: no tearing or untearing/gluing. Next you can talk about rudimentary invariants like things being path connected, if I stay inside the space can I get o every other point. Next, when i remove a point is it still path connected? etc... This gets very strange and hard quickly so you need some new tools to accomplish your goal. So you talk about $\pi_1$ which is not too hard to convince them that they understand. The best space to help them compute the fundamental group of is $\mathbb{R}^2-0$, that is something they can wrap their heads around, but don't mention the group structure. And I talk about how this invariant can detect differences, sometimes, but it doesn't tell you when things are the same.
That takes about 30 minutes or so at least, and it is not about set theory. My recommendation would be to take that model of presentation of a field and use it in the following way. There are certain types of problems in fields, the big problems you know, like classification problems, enumeration, and computation (I am sure there are other big schemes for programs, I just can't think of them, or don't know them). There must be some big program in set theory that you can talk about, and look at early toy problems that people may have tested the theory on. Pick some small examples and use imprecise and soft words that are non technical. Try to draw a picture and don't use mathematical symbols. Maybe you could talk about how $\omega = 1 + \omega \neq \omega + 1$ (where $\omega$ is the first ordinal after all of the finite ones, maybe I have it backwards or wrong). That is kind of cool, and doable... i think.
Anyway, that is what I do when people ask, hope it helps. (I also have a similarly watered down explanation of Spectral Sequences)
Again, apologies for misrepresentations or inaccuracies, no disrespect is meant.