Umbrellas are used primarily as diffusers in photography. They are used either to diffuse light by reflective diffusion, or shoot-through diffusion.
Reflective diffusion: A flash light is fired directly into the underside of a black umbrella with a white or silver lining. The light is reflected back on the scene. The open parabolic shape diffuses the light over a wide area. The light is scattered from the shape by spreading it out.
Shoot through diffusion: This type of umbrella shot uses a white umbrella. The flash is directed into the underside of the white umbrella, but some of the light passes through it to illuminate the scene. This diffusion is therefore direct rather than reflection-diffused.