(Added an epilogue)

I started a job as a TA, and it requires me to take a five sessions workshop about better teaching in which we have to present a 10 minutes lecture (micro-teaching). 

In the last session the two people in charge of the workshop said that we should be able to "explain our research field to most people, or at least those with some academic background, in about three minutes". I argued that it might be possible to give a general idea of a specific field in psychology, history, maybe some engineering, and other fields that deal with concepts most people hear on a daily basis. However, I continued, in mathematics I have to explain to another mathematician a good 30 minutes explanation what is a large cardinal.

I don't see how I can just tell someone "Dealing with very big sizes of infinity that you can't prove their existence within the usual system". Most people are only familiar with one notion of infinity, and the very few (usually physicists and electrical engineering students) that might know there are more than one - will start wondering why it's even interesting. One of the three people who gave a presentation that session, and came from the field of education, asked me what do I study in math. I answered the above, and he said "Okay, so you're trying to describe some absolute sense of reality." to which I simply said "No".

Anyway, after this long and heartbreaking story comes the actual question. I was asked to give my presentation next week. I said I will talk about "What is mathematics" because most people think it's just solving huge and complicated equations all day. I want to give a different (and correct) look on the field in 10 minutes (including some open discussion with the class), and the crowd is beginner grad students from all over the academy (physics, engineering of all kinds, biology, education, et cetera...)

I have absolutely no idea how to proceed from asking them what is math in their opinion, and then telling them that it's [probably] not that. Any suggestions or references?

Addendum:
The due date was this morning, after reading carefully the answers given here, discussing the topic with my office-mates and other colleagues, my advisor and several other mathematicians in my department I have decided to go with the Hilbert's Hotel example after giving a quick opening about the bad PR mathematicians get as people who solve complicated equations filled with integrals and whatnot. I had a class of about 30 people staring at me vacantly most of the 10 minutes, as much as I tried to get them to follow closely. The feedback (after the micro-teaching session the class and the instructors give feedback) was very positive and it seemed that I managed to get the idea through - that our "regular" (read: pre-math education) intuition doesn't apply very well when dealing with infinite things.

I'd like to thank everyone that wrote an answer, a comment or a comment to an answer. I read them all and considered every bit of information that was provided to me, in hope that this question will serve others in the future and that I will be able to take from it more the next time I am asked to explain something non-trivial to the layman.

I have given talks about mathematics to non-mathematicians, for example to a bunch of marketing people. Supplemental: to see an example of a talk of mine that was given to a general audience, see my TEDx talk "Zeroes" (with supplemental material). The talk lasted 15 minutes and it took me about two weeks to prepare.

In my experience the following points are worth noting:


If the audience does not understand you it is all in vain.
You should interact with your audience. Ask them questions, talk to them. A lecture is a boring thing.
Pick one thing and explain it well. The audience will understand that in 10 minutes you cannot explain all of math. The audience will not like you if you rush through a number of things and you don't explain any one of them well. So an introductory sentence of the form "Math is a vast area with many uses, but in these 10 minutes let me show you just one cool idea that mathematicians have come up." is perfectly ok.
A proof of something that seems obvious does not appeal to people. For example, the proof of Kepler's conjecture about sphere packing is a bad example because most people won't see what the fuss is all about. So Kepler's conjecture would be a bad example.
You are not talking to mathematicians. You are not allowed to have definitions, theorems or proofs. You are not allowed to compute anything.
Pictures are your friend. Use lots of pictures whenever possible.
You need not talk about your own work, but pick something you know well.
Do not pick examples that always appear in popular science (Fermat's Last Theorem, the Kepler conjecture, the bridges of Koenigsberg, any of the 1 million dollar problems). Pick something interesting but not widely known.


Here are some ideas I used in the past. I started with a story or an intriguing idea, and ended by explaining which branch of mathematics deals with such ideas. Do not start by saying things like "an important branch of mathematics is geometry, let me show you why". Geometry is obviously not important since all of mathematics has zero importance for your audience. But they like cool ideas. So let them know that math is about cool ideas.


To explain what topology and modern geometry are about, you can talk about the Lebesgue covering dimension. Our universe is three-dimensional. But how can we find this out? Suppose you wake up in the morning and say "what's the dimension of the universe today?" You walk into your bathroom and look at the tiles. There is a point where three of them meet and you say to yourself "yup, the universe is still three-dimensional". Find some tiles in the classroom and show people how always at least three of them meet. Talk about how four of them could also meet, but at least three of them will always meet in a point. In a different universe, say in a plane, the tiles would really be segments and so only two of them would meet. Draw this on a board. Show slides of honeycombs in which three honeycomb cells meet. Show roof tilings in which thee tiles meet, etc. Ask the audience to imagine what happens in four dimensions: what do floor tiles in a bathroom look like there? They must be like our bricks. What is a chunk of space for us is just a wall for them. So if we have a big pile of bricks stacked together, how many will meet at a point? At least four (this will require some help from you)!
To explain knot theory, start by stating that we live in a three-dimensional space because otherwise we could not tie the shoelaces. It is a theorem of topology that knots only exist in three dimensions. You proceed as follows. First you explain that in one or two dimensions you can't make a knot because the shoelace can't cross itself. It can only be a circle. In three dimensions you can have a knot, obviously. In four dimensions every knot can be untied as follows. Imagine the that the fourth dimension is the color of the rope. If two points of the rope are of different color they can cross each other. That is not cheating because in the fourth dimension (color) they're different. So take a knot and color it with the colors of the rainbow so that each point is a different color. Now you can untie the knot simply by pulling it apart in any which way. Crossing points will always be of different colors. Show pictures of knots. Show pictures of knots colored in the color of the rainbow.
Explain infinity in terms of ordinal numbers (cardinals are no good for explaining infinity because people can't imagine $\aleph_1$ and $2^{\aleph_0}$). An ordinal number is like a queue of people who are waiting at a counter (pick an example that everyone hates, in Slovenia this might be a long queue at the local state office). A really, really long queue contains infinitely many people. We can imagine that an infinite queue 1, 2, 3, 4, ... is processed only after the world ends. Discuss the following question: suppose there are already infinitely many people waiting and one more person arrives. Is the queue longer? Some will say yes, some will say no. Then say that an infinite row of the form 1, 2, 3, 4, ... with one extra person at the end is like waiting until the end of the world, and then one more day after that. Now more people will agree that the extra person really does make the queue longer. At this point you can introduce $\omega$ as an ordinal and say that $\omega + 1$ is larger than $\omega$. Invite the audience to invent longer queues. As they do, write down the corresponding ordinals. They will invent $\omega + n$, possibly $\omega + \omega$. Someone will invent $\omega + \omega + \omega + \ldots$, you say this is a bit imprecise and suggest that we write $\omega \cdot \omega$ instead. You are at $\omega^2$. Go on as far as your audience can take it (usually somewhere below $\epsilon_0$). Pictures: embed countable ordinals on the real line to show infinite queues of infinite queues of infinite queues...

