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.
