How to explain to the layperson what mathematics is, why it's important, and why it's interesting 
A mathematician walks into a party.

No, this is not the beginning of another joke, nor of a graph theory problem, but rather the beginning of a frequent and often frustrating real-life situation. Somebody asks you: 

"So, what do you do?"

What do you answer? "I am a mathematician / math student / math enthusiast," whichever applies. Then, two things can happen: 


*

*they are satisfied with this answer, which, almost all of the time, means that they are satisfied with their own misconception of what you do. They perhaps imagine that your main tool is a TI-83 calculator, that you spend all day crunching interest rates or solving quadratic equations using that formula they also memorized and forgot long ago. In the best of cases, they believe that your work consists in doing a bunch of calculations with a bunch of numbers, and that it must be incredibly boring. (The only upshot - they usually believe that "you must be smart".)

*if you are lucky, they ask you what you really do, and then you get a shot at explaining.


I'm not sure which of these two situations I find the hardest to deal with, although I have dealt with both countless times. I've come to think that mathematicians are the most misunderstood professionals in the world. Biologists, physicists, writers, sociologists, and maybe even philosophers all have an easier time explaining what they do. Of course, there are popular misconceptions about them also. People might believe that biologists study cells under the microscope, or that physicists study very small things with lasers and very big things with telescopes, or that writers write and drink, or that philosophers study Aristotle. Although none of these beliefs is correct, none seems so far removed from the truth as the belief that mathematicians calculate interest rates with a TI-83 calculator.
Most of the time, I give up before even trying. And those rare times when I do try to give my interlocutor a glimpse of what mathematics is about, I fail horribly. Could it be that the beauty of mathematics is impossible to communicate, except to those people who have already had a taste of it? It might explain why so many mathematicians seem to marry each other - perhaps they feel that no other person could possibly understand them!
So my question is - how do you explain to people what mathematics is? And how do you convey the idea that it is absolutely fascinating and mysterious and that it is one of the greatest human enterprises - all in five minutes? My experience has been that going into any amount of detail - by which I mean uttering a single strange word - is a horrible idea. I'd really love to know how others deal with this.
Thank you all in advance! :-)
Addendum: I am disappointed that this question was closed after less than a day, especially considering that in such a short time it was viewed over 500 times. Obviously this question strikes a chord with some people, and since it is well-posed question which admits real answers, I think it should be reopened. What is the soft-question tag for, if not for a question like this? Why have similar questions not faced the same fate at MO? Anyways, no bitterness - thanks to all of you who shared your insight! (And please vote to reopen if you can. :) )
 A: As an amateur I can possibly not really contribute here. But if I'd extrapolate my fiddlings in number-theory to be contents of my study/profession, I'd simply say:      
"the engineer may compute the parameters of the construction of a bridge by some stiffness and tension formula - but we mathematicians are the ones who develop, provide that formulae and prove, how far a specific formula is valid - what means "working at all".
The computer specialist at Google may build some awesome powerful search-engine for the whole internet, but we mathematicians are the pioneers in the field of numbers: those who find the matrix-formulae to which that engineer relates, and evaluate their properties, benefits, drawbacks and even their limitations: the possible circumstances where they begin not to work... such that they might be used in widest generality, not only for the construction of a bridge ... " 
And, to extend to some emotion, I'd add (because I really like math and especially number-theory) something like "... to find such formula is like finding or to create some pattern in music, in painting - partial in my work is great pleasure of finding intellectual elegance and and beauty: you can do some football/soccer by hard work but also by elegance and creativity. As far as my study/job has niches for creativity, communication, exposition and contemplation of the work one can enjoy also elegance and beauty".
A: "It's hard to explain because mathematicians try to answer questions so involved and difficult that it almost always takes many years of study to even understand the questions.  We try to answer them, and prove that our answers are correct.  Usually the questions aren't even motivated by anything in the 'real world' - one answer leads to new questions, which leads to new answers, etc.  But it's all surprisingly practical in the end.  Eventually someone comes up with a practical problem that is answered by the previously impractical mathematics."
Or, just say:
"I mostly just sit around and think about math, and try to come up with some new stuff.  I get paid either way."  
A: We have all heard of the famous anecdote about David Hilbert quipping when he heard one of his students dropped out for poetry: "He never had the imagination to be a mathematician anyway." 
Mathematics requires supreme imagination of abstract lines, shapes and quantities. Yes, mathematics is stoic art, can be dry and monotonous, and maintains a cold, formal character but once you get past the syntax and semantics, and the numbers come to life in holistic picture, structures, concepts, and ultimately philosophy. 
Mathematics is no more a study of computation than a score is that of meaningless musical notation. Part of the beauty lies in paucity and simplicity. Consider the diagonal lemma. Once you 'get it' like a good joke or art, you can then only appreciate the 'wit' in the proof.
Although there has been a huge influx of popular accounts about mathematics, does mathematics, by itself, need any advertisement just like a ventriloquist or snake charmer needs any ancillary halo? Although the subject may enjoy a hierarchy such as governing laws of quantum mechanics or postulating about certain uncertainties such as Church-Turing thesis, the kernel, without any gloss can only be appreciated to the fullest in total immersion.
Sometimes it is the mathematicians that makes the subject colorful or 'sexy'. They breathe life into the animate concepts. Kleene was a mountaineer and canoeer. Shimura's poetic soul is reflected in his book on Imari porcelain or Eilenberg through eastern art collection. There were pacifist like Bertrand Russell and war hero like Emile Borel. Just like the subject, the participant becomes part of it and becomes paradoxical. Godel, a man of supreme intellect and reason, became paranoid at the end. Turing meanwhile to certain extent treated or thought himself as a machine. There were polyglots and musicians. Fencers and cartographers, clockmaker, gamblers, astrologers and astronomers. Tarski and Erdos abused substance while there were mystic like Cantor and Brouwer. There is juggler like Graham, and of course insanity as the muse for creativity for countless others. Tragedies like men committing suicide out of frustration or post-war Era exist, while there child prodigies like Galois and von Neumann.
Of course, one cannot deny the accolades and honors bestowed on them. Some are commemorated on Eiffel Tower, some Nobel laureates or Abel recipients, or have asteroids named after them.
But to naively assume, the men live for these prizes would be to miss the point. After one receives doctorate in mathematics, the journey merely begins and gets steeper. In the end, it's not for honor, prestige, fame, riches or alpha-male like anecdotes. Rather,
Ars gratia artis
A: My two cents.
In general, I think Mathematics is best explained to non-mathematicians by examples (and to mathematicians by counterexamples, LOL).
According to Raskolnikov, you should:

start with an example of the kind of mathematics that you find absolutely wonderful and that you can convey in words. Even if it's not related to your research directly.

When I read this passage, I cannot help remembering a quote (from Goethe, if I'm not mistaken):

Beauty is in the eye of the beholder

and observing that wonderfulness resembles beauty in so many aspects. I mean: wonderfulness is not a general, nor the only criterion to catch people's attenction.
In many situations usefulness is catchier than wonderfulness, e.g. when you talk to musicians, engineers, botanists or ethologists. For example, there is a mathematical link between the reasons why cats curl up to sleep, why many fruits are spherical, why columns' sections are discs or why drums are (mostly) round: the isoperimetric inequality.
A: I am not a mathematician, and I don't care what kind of mathematics someone I just met at a party finds absolutely wonderful - and I don't think anyone else will either.
"I am a mathematician / math student / math enthusiast" doesn't actually answer the question: "So, what do you do?".
In fact, what that question is actually asking is: "How do you make money?".
People already have an idea as to how accountants, programmers, burger-flippers, etc make money. The layman really doesn't have any idea how a mathematician makes money, which is why they may make the assumption that you just sit at a desk doing calculations all day, everyday; probably for an accountant or statistician.
So, to answer their question just tell them how you make money, better yet, let them infer the reason why someone would pay money for them to do what they do.
Examples:


*

*A programmer: "I'm a web-programmer, I make websites similar to facebook."

*An engineer for NASA: "I'm an aerospace-engineer, I help NASA [do whatever a NASA engineer does]."

*A musician: "I'm a drummer, I play the drums for Lynyrd Skynyrd."

A: Why not start with an example of the kind of mathematics that you find absolutely wonderful and that you can convey in words. Even if it's not related to your research directly. 
One I find fascinating and that every thinking person must have pondered at least once in his life is the infinite. Now, who else but mathematicians can tell us the best about it? You can start with a kind of riddle. Are there more even numbers than odd numbers? Or are there more even numbers than numbers? You immediately come into issues of what it means to count and into set theory. Things that seem trivial and taken for granted by most non-mathematicians, but are really not.
If you work in statistics, you could talk about famous probabililty/statistics paradoxes. If your field is analysis, there are many weird functions that can be constructed that go against intuition but that you still can more or less convey in words or drawings what they are. If your field is algebra, you can talk about symmetry (think Escher drawings for instance). There's always something. 
A: What do mathematicians do ?  Mathematicians do mathematics.  What is mathematics ?  It is an appendage of natural language that is a bit more precise than natural language and any human knowledge that is not based on intuition or experience or custom can often be abstracted into mathematics and surprisingly, in this abstract world, one can see the pure essence of the knowledge captured (or sometimes, but rarely, jumbled).  Mathematics is what offers physics, theoretical chemistry, engineering, medicine, computing, business, logistics, networks, ... thought-itself to have a language beyond natural language with which to express itself and understand itself especially in regions when natural language does not suffice to elucidate.  Mathematics is the language of abstract knowledge.  Mathematicians are the grammarians (and poets) of this language.
