There are all sorts of assumptions behind the question, and answers, such as that the solution of famous problems is the test of progress in mathematics. At my first international conference in 1964 I met Stanislaw Ulam, and he mentioned to me: "A young person may think the most ambitious thing to do is to tackle some famous problem or conjecture. But that might distract that person from developing the kind of mathematics most appropriate to them."

In the 1980s I gave a talk to teachers and children on "How mathematics gets into knots" and mentioned prime knots and prime numbers. After the talk, a teacher came up to me and said: "That is the first time anyone in my career has used the word "analogy" in relation to mathematics." I find that tragic!
We have to be careful not to fall into: "They ask for bread and we give them stones."

The notion of fractal is well known to the public, but how many mathematics courses give a simple account of the Hausdorff metric, and let students see some of the mathematics behind the fractal notion.

Other scientists would like to know what is new in mathematics, in terms of concepts and ideas. My talk to a Conference on Theoretical Neuroscience in 2003, was well received. it included an email analogy for colimits, as well as ideas on higher dimensional algebra. One participant told me: " That was the first time I had heard a seminar by a mathematician which made any sense!" So this time I managed to get it right!

First year main math courses at University should contain something which excites the imagination. (I am told Physics courses usually have something on current research.) That Euclidean geometry has been out of most syllabi makes this harder, especially to get over the idea of proof. Is it too harsh to say that courses on "Proof" are about how to write clear proofs of boring things? It is good to show proofs of otherwise not so believable things.

In the 20th century, a main contributor to the unity of mathematics has been Category Theory; I feel this is a high order mathematical achievement! See the article Analogy and Comparison for a discussion of analogy in this context.

A simple talk to the first year on cubes of dimension $0$ to $5$, and how to count faces of various dimensions, awakened the interest of a student, who later went on to a PhD. (This was also a talk I have given to 13 year olds: they end up by counting the $2$-dimensional faces of a $5$-dimensional cube.)

There is also a lot to say about the contribution of mathematics over the millennia to science and culture. See an article on Mathematics in Context.

Edit: January 12, 2014 : I would like to add that since the general public are often familiar with the words **fractal** and **chaos**, it is sad if undergraduates in mathematics are not given some idea of the rigorous mathematics behind these notions, in particular for fractals that is the notion of Hausdorff metric. I have given a light hearted course on this in a second year course on analysis, without proving the completeness theorem, but explaining what it means, and with exercises on calculating the Hausdorff distance between subsets of the plane. I also asked them to do a short project on "The importance of fractals" using the web to get evidence, and also encouraged use of fractal computer programs.

The notion of "chaos" is also important in view of the financial situation, and the everyday notion of weather and climate!