TITLE: Cocktail party math
QUESTION [58 upvotes]: Ok, hotshots. You're at a party, and you're chatting with some non-mathematicians. You tell them that you're a mathematician, and then they ask you to elaborate a bit on what you study, or they ask you to explain why you like math so much.

What are some engaging ways to do this in general? What are some nice elementary results, accessible to people with any mathematical background or lack thereof, that can be used to illustrate why math is interesting, and its depth, breadth, and beauty? Note the scenario: you're at a party so it should be a relatively quick and snappy kind of thing that doesn't require a blackboard or paper to explain.
Easy Mode: How do you explain what your particular sub-field of math is about in an accurate but still understandable and engaging way? Hard Mode: Assuming you work in a less applied area, how can you do this without mentioning any real-world applications? Again, note the scenario.

This question is inspired by this one, in particular Anton's answer.

REPLY [2 votes]: When trying to talk about specific results, I really like talking about Cantor's Theorem (or at least, the special case of $2^{\aleph_0}$), and then, if they're willing to accept that, talk about ordinals a bit. If the audience isn't taking it, I'll generally talk about some arbitrary graph problem that comes to my head.
But when asked, I typically try to approach it from the more philosophic perspective of "what mathematicians do"-- study abstract structure. (Or at least, that's the approach I take to math), and try to explain what that means, providing some examples here and there. The definition of group shows up a lot, since there are easy to understand examples of groups. I also view myself as a bit of an artist, so I tend to use analogies that deal with music and painting.