A mirror is an object that reflects an image / light / photons. There are mirrors with different reflectiveness and different construction. In the video the prof says they are using a "highly reflective nanostructured mirror", so it's obviously not exactly your silvered glass bathroom variety mirror, but in principle it's the same: a surface that reflects almost all photons, and absorbs or transmits almost none.

The box is also not necessarily like your average shoe box, I'd guess it's much smaller (millimeters), show it's not shown on the video directly, up close.