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I have read the other answers here, and I think they all offer very useful insights. I'll adopt some of them myself, thank-you all! I'd like to offer one more, if I may...

Whilst my own approach will be very culturally dependent and will not work everywhere, I can say that it works perfectly well in Australia:

I use humour. Self-deprecating humour more often than not. (It's a personal favourite).

I am an organization theorist. Practically, that means I study management and organizations. Academically, 'management' covers a range of sub-fields from soft 'people' things like organization behaviour (which is, in effect, psychology) to hard things like the ways that organizations are structured (and structure themselves), and strategy and such, and a range of other things in between those two extremes along a kind of soft-to-hard continuum.

I have just submitted a doctoral thesis about organization theory. In it I examine the organization structures and power relationships among a group of people who make up a large Open Source Software project. The nature of 'employment' in such an organization means that power relationships are largely inverted as compared to those in a conventional organization, and so I examine those relationships, along with psycho-social and organization-theoretical notions of 'motive' and 'self' and power and resistance and such in the hope of informing management practice in the 'real' world.

That's a hell of a mouthful to explain, and I've already vastly simplified it for this answer. There's a real risk in rendering the conversation at a dead-end from the outset if I start like that. It wasn't until a month or so after I completed that I realized that what I had done was:

"I wrote a book about people fighting among themselves on the Internet".

and that makes me

"An expert in fighting with strangers on the Internet".

This approach works pretty well for me. If a person really wants to make polite untechnical conversation, this tells them what I'm up to, and lets them off-the-hook insofar as being dragged into incomprehensible technicalities goes. If I've dumbed it down TOO much, and my audience (of one) actually does have the capacity to understand more, my flippant approach is self-deprecating, so I'm insulting myself, not them. Again, this is very culturally dependent...

In Australia we have the notion of the "tall poppy syndrome". Analogically, it visualises a field of poppies in which one flower grows taller than the rest, and because it's "head is sticking up", it is at greater risk of being cut down than others. Socially, Australians are uncomfortable with self-promotion and will seek to 'cut down' any 'poppies' that stick their heads up. It is similar to the Filipino notion of "crabs in a bucket" wherein if you put a single crab in a bucket you must put a lid on top to keep the crab inside, but if you have two or more crabs, you don't need a lid because they'll continually pull each other down.

There is another idea, one that I suspect extends further than my own national culture, and that is the broad perception that much funded academic research is meaningless, trivial and wasteful of public funding. Layfolks don't understand the notion of "standing on the shoulders of giants" as the means of building reliability and veracity as we do, so explaining that I spent ten government-funded years watching people argue on the Internet will likely be met with a kind of unspoken derision. I think that folks recognize that research is valuable and useful, but they don't see the long-term connections between individual studies and societal benefit. By describing my work in humourous terms, I allude to that social concern in a way that makes no attempt to defend against it and keeps me firmly in the socially-requisite 'bucket' until my audience chooses to life the lid.

So, explaining my work as "fighting on the Internet" is culturally appropriate here. It's simple, it's self-deprecating (a precise opposite for self-promotion) and, believe it or not, it's accurate. So if my audience DOES want to know more, I can develop the facetious explanation into a more technical explanation about how studying people fighting on the Internet genuinely informs those other theories that I raised above, and how my research will directly inform management practice so as to make managers more successful, staff happier and companies more profitable.

It works for me.

I adopted the approach, in part, after observing an evolutionary biologist whom I admire, Prof. Rob Brooks of the University of New South Wales in Sydney, Australia, regularly introduce himself - often in quite solemn situations and in senior, important and elite company - by saying "it is my job to spend a lot of time thinking about sex". It always makes the more 'proper' individuals ever-so-slightly uncomfortable, but it amuses everyone, and it is accurate.

In both cases, if the inquirer is happy with that answer, they'll leave it there. If they want more, if they want to talk about sex or about fighting strangers, then these flippant and slightly facetious responses are genuine, accurate and valid, and can be developed into more properly technical explanations of our work.

I'd like to add that I think it is important for all scholars, or anyone who does deeply technical work, to be able to explain what it is that we do to laypeople. As @chris-f-carroll pseudo-attributes to Feynman in his comment elsewhere on this page, though perhaps not for the same reason, it is important to be able to explain oneself to non-technical audiences. At some point, you'll want to explain the value of your work to your Mum or your partner or a senior manager who controls funding that you need. For that reason, we encourage research students at all levels to keep thinking about how to explain their work: we set "explain your 80,000 word thesis thoroughly and accurately in 150 words" tasks, ask them to speak for no more than 10-15 minutes at bi-annual progress reports (and to take questions from their audience), and encourage participation in "Three Minute Thesis" competitions.

A final point I will make is that those of us who work at deeply technical things are well used to being misunderstood - or simply not understood - by laypeople. Very occasionally, we meet individuals who do understand what we're talking about and find ourselves having inadvertently "dumbed it down" too much for a particular audience. It feels a little embarrassing when that occurs, but the situation is easily recovered by a brief and honest apology. The other person always understands.

Thanks for the question and the opportunity to reply. I hope my perspective helps somehow.

Geoff