I'm interested in wit. In my opinion, it's the clearest and probably best way to show one's level of intelligence. However, after looking over the work of greatest wits of history, I've noticed some patterns. These can all be summed up into one point: wit is word-based.
The usual witticism will convey two elements. First, either a deep philosophical statement, or an insult of some sort. Second, a completely unrelated play on words.
For example, the great wit Oscar Wilde has been attributed the quote, "There is only one thing in the world worse than being talked about, and that is not being talked about." The philosophical point is that Wilde would rather be infamous then unknown, and the wordplay is that he used a seeming contradiction of words to say so.
As another example, Churchill has been said to have been invited to a play with "Bring a friend...if you have one" and replying with something alone the lines of "I have a previous engagement, but I'll gladly attend your second night...if you have one." Or maybe it was someone else who said it. I don't care, since it's beside the point. The joke follows the same pattern as most other wit: It has a insightful or insulting comment (and this time it's an insult) made funny by wordplay (parallel construction this time).
I've always thought that wordplay was a bit of a shallow way to be witty, since it's usually pretty easy. But after looking through the different wits, from Whistler to Twain to Marx (just kidding, I meant Marx), wordplay was just about the only thing they could use.
As a corollary, this explains why puns are disliked so much. Since they try to play on words, but tend to be unimpressive, they are the one form of joke that has the highest aspirations but goes the least towards reaching them. I still like them anyway. Probably because they're the underdog.