Umbrellas are available in two basic types, reflective and shoot through.
Shoot through umbrellas are capable of producing very soft lighting, because they can be placed very close to the subject if required. Their downside is that the light is fairly uncontrolled, and about 40% of the light is reflected backwards, which can cause lens flare as well as adding to the loss of lighting control when light bounce off of nearby walls.
Shoot through umbrellas are a perfect tool for creating on-axis fill light.
Reflective umbrellas cannot produce such a soft light as a shoot through umbrella because the flash head is positioned between the umbrella and the subject, which prevents the umbrella from being really close to the subject. However, there is far less light spill than from shoot through umbrellas, and the light is more controlled.
Reflective umbrellas are available in white, silver and gold. White produces the softest light, silver produces a harder, more specular light that can add drama and impact, and gold produces a similar result to silver, but of course with a gold tint.
Reflective umbrellas can, if required, be partially collapsed to create different effects, adding to their versatility.