Diffuse reflector is a mirror with big scattering of the light. An example of usage: a white fabric umbrella as a reflector for a lamp in photography to make shadows softer.
Spectrally flat reflector is a reflector with constant reflectance versus wavelength.
In context of fiber optic sensors I can imagine a micro fluid cell with a diffusive reflector in front of a fiber ends for spectroscopy of the fluid. Diffusive reflector has an advantage over a mirror that it's less sensitive to alignments and e.g. refractive index gradient in the fluid.
In the geometrical acceptation, a perfect diffuse reflector is a material that has the property of reflecting the light it receives according to a lambertian mode, i.e uniformly in the whole half sphere over its surface. Such a reflector is to be opposed to a specular one, such as mirrors that reflects light in a preferred direction, with respect to the illumination conditions.
Diffuse standards are often made from pressed Teflon powder and can be purchased for calibration purposes. They are not perfectly Lambertian but that's the best thing we can do at the present time for diffusing surfaces.
Amir, What Guillaum mentioned is material called spectralon coated inside the integrating sphere wall, of which diffuse reflectivity is high and nearly constant from UV to near IR as shown in file. Test target with spectral on is available from Edmund Optics.