I enjoy sharing Math with non-mathematicians. Here are the slides I used in a talk aimed at non-mathematicians describing what Commutative Algebra is. I was also a member of the UVa Math Ambassadors, and I volunteered for Wolverine Pathways.
Among other activities, I co-edited a book that includes 32 short articles written by 33 young mathematicians, each article explaining a different topic in Math to a broader audience, assuming only high-school level math. However, the book is written in portuguese - but if you happen to speak the language, check out the links below! A rough translation of the title would be something like Numbers, surgeries and tie knots.
I also wrote one of the articles in the book, explaining some basic algebra (groups, rings, etc) and Galois theory. The title roughly translates to From play-doh to 5th degree equations.
How many tie knots are there? What are Penrose tilings? What are hyperreal numbers? How about p-adic numbers? Is it really impossible to find an algebraic expression to solve general 5th degree equations? Is it possible to use message cards to simulate a computer? What is the connection between linear algebra and the hydrogen atom? How do we compute derivatives of a function with discontinuities? How does one use knots and surgeries to understand 3 dimensional sets living in 4 or 5 dimensional spaces? Can Math say anything about population growth and competition between species? Or about medical diagnosis? Or on how to control satellite orbits? These are some of the questions answered in this book.