How do you explain your pure maths PhD?
This questions will vary greatly depending on your research area. Many people in applied fields will have no more issue than any other scientist, while those working in, say, algebraic geometry tend to have a tougher road. That said, here are a few principles I have found helpful:

1) Find the smallest question that captures the key idea. You don't have to explain your particular problem, so much as give a flavor of the sort of question you work on. For example, in Schubert calculus, everyone always leads with the question "Here are four lines in space. How many lines intersect all four of them?", even for other mathematicians! You can then allude to higher dimensions, broadly say how your work ties in (or just assert that it does!), and so on.

2) Return to the specific. Use vivid metaphors as a way to overcome abstraction. For example, I often describe a permutation as a row of line dances, and a simple transposition as two people dosey-do-ing to swap places. Now I can talk about the dancer's motivation as motivating my questions, and it somehow seems less arbitrary. Providing something tangible to visualize helps a great deal. This is completely at odds with the usual mode of mathematical communication, where we wish to abstract away every specificity.

3) Inject narrative. Talk about the history. Mention peoples names, and say what they did. Give a scope of the human endeavor that is (a) mathematics, (b) your field and (c) your specific problem. "Littlewood and Richardson came up with a rule for multiplying these polynomials (Schur functions) in the 1930's, and said the proof follows from 'simple combinatorics', which they thought beneath them to do. Forty years later, Schutzenberger finally figured out how to do the simple stuff, after many failed attempts by famous mathematicians, some of which were published!" It would here be appropriate to mention some details about Schutzenberger's fascinating life.

Edit: Explain why you care! Talk about how you came to the problem, your motivations (beyond glory) for solving it, why you think it's worthwhile.

4) Create opportunities for dialogue. Obviously this only applies to someone who is genuinely curious, as opposed to being polite. If you provide the over-arching perspective (1) and provide a specific and familiar framework (2), your listener will have a framework they can use to start asking questions. Metaphors will be abused, and their limits exposed, but that's okay. At the end of it all, someone might call you a "math detective".

5) Be willing to sacrifice a little (or a lot) of accuracy. It's okay to describe the overall thrust of your area, rather than the particular question that you work on. It's okay to give a wishy-washy description of something that glosses over many complications. Despite working in the field that most prizes accuracy, mathematicians regularly gloss over subtle issues with each other. The standard should be much lower when dealing with non-mathematicians. Seriously, it's okay!

6) Steal shamelessly from others in your field. Read popular accounts of your area, or ask other people how they try to describe things. If you work in the Langlands program, read Ed Frenkel's book and see how he tackles this challenge. Look at the "What is a..." series in Notices of the AMS. In general, people seem averse to doing this across academia, but everyone benefits if you can use the best exposition, regardless of whether or not it is original to you.