When asked what a group is, my favorite examples to start with are the symmetries of a regular polyhedron. After some thought it can be seen that each such group is generated by reflections. There is an interesting correspondence between groups generated by reflections, the arrangement of hyperplanes fixed by their reflections, and the braid groups arising from the complement of said hyperplanes. In this talk I will discuss these groups and also some recent results about one particular affine extension of a complex reflection group.
See some associated pictures here and here.