How does the work of a pure mathematician impact society? First, I will explain my situation.
In my University most of the careers are doing videos to explain what we do and try to attract more people to our careers.
I am in a really bad position, because the people who are in charge of the video want me to explain what a pure mathematician does and how it helps society. They want practical examples, and maybe naming some companies that work with pure mathematicians, and what they do in those companies. All this in only 5 or 10 minutes, so I think that the best I can do is give an example.
Another reason that I am in a bad position: In my University we have the career "Mathematical engineering" and they do mostly applications and some research in numerical analysis and optimization. (*)
I know that pure mathematics is increasing its importance in society every year. 
Many people in my country think that mathematics has stagnated over time and now only engineers develop science. 
I think that the most practical thing I can do is give some examples of what we are doing with mathematics today (since 2000).
If some of you can help me, I need the following: 


*

*A subject in mathematics that does not appear in (*). Preferably dynamical systems, logic, algebraic geometry, functional analysis, p-adic analysis or partial differential equations.

*A research topic in that subject.

*Practical applications of that research and the institution that made the application.
Extra 1. If you know an institution (not a University) that contracts with pure mathematicians and you know what they do there, please tell me also.
Extra 2. If you have a very good short phrase explaining "what a mathematician does" or "how mathematics helps society" I will appreciate it too.
Thanks in advance.
 A: Compressive sensing and time frequency analysis might fit your bill. Suppose you have a signal (say a recording of someone's speech) that you want to transmit. In order to do this, you break the signal into a discrete set of pieces in such a way that from the pieces alone you can reconstruct the entire signal. Then you can send the discrete pieces of information and the receiver can reconstruct the signal.
Abstractly, this amounts to asking "When can you reconstruct a function $f \in L^2(\mathbb{R}^d)$ if all you know are the values of $f$ on some lattice, or other discrete set?" You need some conditions on $f,$ and some conditions on the point set, but surprisingly such a reconstruction is possible. Although I have largely billed it as an applied problem, there are many rich connections to areas of pure math, starting with Fourier and time frequency analysis, but leading into dynamics and operator algebras as well. 
A: Personally I prefer the term "theoretical" mathematics rather than "pure."  But just to give you an answer to Extra 1: Microsoft Research employs mathematicians trained in theoretical fields: for example topologists (e.g. Mike Freedman, Zhenghan Wang, Kevin Walker) at Microsoft Station Q in Santa Barbara are collaborating with physicists in an attempt to realize topological quantum computers.  
A: Linear algebra might be useful here. A classical research topic in linear algebra is the spectrum of matrices with non-negative entries (a matrix could be described maybe as an abstraction of geometric operations like rotations, reflections etc). Google's PageRank algorithm has been widely described, and is heavily based on these kind of aspects of linear algebra. 
A: Maybe not quite what you asked for, but the report
Measuring the Economic Benefits of Mathematical Science Research in the UK (2.5MB)
http://www.cms.ac.uk/files/Submissions/article_EconomicBenefits.pdf
may be relevant. There are a few good pure maths examples in there.
A: See: http://www.whydomath.org/
Plenty of pure mathematics keeps turning into "applied" mathematics as time goes by.
That is the gist of how math impacts society. I would stay away from the more philosophical answers such as "Math helps you to think logically" etc., since your audience would probably not appreciate that. 
Some of the more apparent uses of high level mathematics:
1). Theoretical advances in control theory that lets us fly UAVs, supersonic planes etc: 
Lockheed Martin/Boeing/etc. often employ mathematicians (pure and applied) to work on control aspects
2). Theoretical advances in communication theory that has revolutionized the telecom business:
Companies such as Qualcomm,Samsung etc. employ many PhDs who work in theory.
3). Pure Mathematicians such Edward Belbruno (NASA/Princeton) helped devise new ways of interplanetary travel using exotic properties of the three-body problem.
A: Of course there are many companies hiring pure mathematicians. But they do not work as pure mathematicians, they become programmers, consultants, or statisticians… That is the truth and prospective students deserve to know it.
A: Here are two recent speakers from our departments lecture series on applied mathematics (the Wing Lectures at U. Rochester). We've had a lot of great lecturers, but these two stick out to me as having an impact on society.
Adrien Treuille -- he works in computer graphics, and his research purely in this areas includes algorithms realistic modeling of crowds, and real time fluid mechanics (e.g. with basically no delay, they can add digital trails of flames behind a race cars on TV). Also, he can construct initial data to make (digital) smoke form specified shapes at specified times.
But, what's even cooler, is that he's collaborated with biologists on an interactive game (called FoldIt) for finding the optimal conformation of proteins. Players get points for moving the protein into better conformations. Humans are way better at this than computers, and they've actually published papers based on conformations discovered by players; in one case, the answer they found had eluded scientists working in the field for at least 10 years! They call this "crowd source science". They've also created another game for engineering shapes with RNA, and the players of that game have discovered things as well.
Gunnar Carlsson -- he is an algebraic topologist who originally worked in K-theory, but shifted to using algebraic topology to understand data. Specifically he was one of (the?) pioneers of persistent homology, which is a way of using topology to understand data. The really great thing about it is that it discovers structure for you--instead of fitting the data to a model, persistent homology discovers the model for you. For instance, they used PH to find the most commonly occurring 9-bit patterns in black and white images, which in theory would allow better image compression that JPEG (but in practical, JPEG is highly optimized, so it would take a lot of work to benefit from this discovery). In another example, they analyzed genomic information from cancer patients and linked it up with survival information; PH discovered the threshold for the expression of a certain gene at which survival drops significantly.
Besides discovering structure in data, PH is also able to incorporate data collected at different times and from different sources (perhaps a reflection that the method is "metric free" --I'm not sure about that). A good reference to check is their paper in Nature.
Also, Tony DeRose from Pixar gave a really great talk on the methods they've developed in computer graphics. I don't remember the details so well (maybe because I was distracted by the entertaining presentation :) so here's a link to a good article.
A: Turing's thesis and the development of the modern computer. John von Neumann's contribution to the war effort (and that of Turing and Hilton, for example). Fractals and modern computer imaging. Wavelets and data compression. Topological data analysis. x-ray diffraction. Number theory and cryptography. Linear programming, control theory, telecommunications, and quadro-copters. Differential geometry and automotive design. Maybe it is currently out of favor, but the NSA is the largest employer of mathematicians in the world. To expand upon that thought, the general data analysis that occurs in government and industry that helps determine buying habits, consumer behavior, and other large scale human behaviors. 
A: Clifford algebras have a wide range of applications inside and outside of Mathematics, including Differential Geometry, Computer Vision, Robotics, Theoretical Physics, Computer Science. See for instance: http://www.amazon.com/Geometric-Computing-Clifford-Algebras-Applications/dp/3642074421
A: The National Academies of Sciences recent report,
The Mathematical Sciences in 2025 (NAS link here),
has a chapter entitled "Connections Between the Mathematical Sciences and Other Fields,"
and details many such connections, as indicated below:
          
The whole report could be useful to you. It can be read online free.
A: General Relativity is always one of my primary examples how really advanced math is important in daily life: Who would have though in 19th century that Differential geometry and Mikowski Geometry would be important to describe our universe? However, GPS does not work without this.
And in the moment, things get further, as the Standard model of particle physics is a differential geometric model. 
A: One example is Latin squares and its application in design experiment. You can search in the web for more information about it and its application in everywhere.
Another example is Line Group. The line groups describe the symmetry of quasi-one dimensional crystals. For more details, you can see the book:
"Line Groups in Physics".
Also, You can see the very interesting book with name:
" I' explosion des Mathematiques " that is in French language.
Finally, I give a sweet example that maybe your audiences love it. The number of divisors of a small number is very important. it is a pure number theoretic problem. But, suppose a company want to packing its production like as chocolate or  biscuit. They prefer to use number 4, 6 or 12. Why? for better sharing the chocolates or biscuits  in a package among some friends.
A: An applicable area of research that relates to PDE is inverse problems, for example the Calderon problem (see e.g. http://www.math.washington.edu/~gunther/publications/Papers/calderoniprevised.pdf ).
Applications include medical imaging (electrical impedance tomography), discovering oil under the sea, and finding cracks in concrete blocks.
Fields of mathematics related to inverse problems include PDE, microlocal analysis, integral transforms and Bayesian statistics. There is also numerical analysis.
Medical imaging and imaging concrete blocks are under current research at least in universities of Helsinki and eastern Finland, respectively.
A: I'm not a mathematician (more of a general science geek) but I ran across this question by chance, and I feel that mathematician John Nash deserves a mention.
He had important direct and indirect positive impact on society, both through his mathematical models of game theory and the role of money in society, as well as aspects of evolutionary psychology, where he personally demonstrated that an individual with severe mental illness (paranoid schizophrenic) can function and make useful contributions to society.
A: From What is Pure Mathematics? :

Finance and cryptography are current examples of areas to which pure
  mathematics is applied in significant ways.

While Pure Mathematics may not have immediate, direct applications, there are those that graduate with Pure Math degrees that do various kind of work given the skills developed in completed the courses, at least at the U. of Waterloo where I studied Combinatorics & Optimization and Computer Science.
