Here is a youtube version
of the slides of my first lecture on the course MathE-320 of spring 2015 Every 2 hour lecture
features a slide show like that:

This course gave an overview over some topics of
mathematics. In the first lecture, an overview over all mathematics
was attempted [PDF]. There are
many books which give better overviews. One is
"Timothy Gowers: Mathematics, A very short introduction."
Of course, there are also more and more online resources.
Wikipedia for
example defines Mathematics as the science of quantity, structure,
space, and change.

The question, "what is mathematics?"
is answered in many different ways. In the lecture of fall 2015, I gave a few examples.
When designing the course in 2009, I tried the definition

Mathematics is the science of structure.

The shortness of the definition makes it more general. Any list of particular things is
too narrow. Galileo wrote that "The grand book, the universe, is written in the
language of mathematics". And added that
"without which it is humanly impossible to understand a single word of it;
without these, one is wandering about in a dark labyrinth"
Here are some additional ones:
Beutelspacher (a book in German with the title "In Mathe war ich immer schlecht")
defined (translate): "Mathematik is the attempt to discover logical connections".
I find the "attempt" part not satisfying. It is too humble. Mathematics does discover
logical connections, not only attempt it. But there is more to it than just logical
connections.

Albert Einstein: "Pure mathematics is, in its way, the poetry of logical ideas."

James Sylvester: "Mathematics is the music of reason".

Georg Cantor: "The essence of mathematics lies in its freedom."

Here the least favorite, even so expressed by giants in the field. It is almost
certain however that all of these quotes were meant to be sarcastic,
similar to other more humorous expressions given in my lecture given this fall.
Here are the three:

Bertrand Russell: "Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true."

David Hilbert: "Mathematics is a game played according to certain simple rules with meaningless marks on paper."

Henri Poincaré: "Mathematics is the art of giving the same name to different things."