(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.