Transparent objects are objects which can be seen through, are usually made of prims that have had their transparency altered, or have a transparent texture. Avatars can not be made transparent, but they can be made invisible through use of a "hack" (a special texture that used to be used for objects that were loading).
An object can not be transparent and shiny at the same time. However, in 1.9.1 they could be, before most of the new shader code was removed--this means in the future transparent + shiny may be possible.
Transparency is a property that can be applied to objects through the edit window > Textures tab or by script. A transparent texture, or the built-in transparency level, can be used to make an object transparent.