The Shadow is a very broad and abstract term that seems to be used in reference to the darkening of the minds of people as a result of the Marring of Arda. It was usually applied to societies that were under the sway of the corrupting power of Morgoth or Sauron; such groups were said to be "under the Shadow" or to have "fallen under the Shadow". An example of this usage is that of the realm of Númenor; 2,000 years after its founding "the Shadow fell upon" it when its people became restless and began to envy the immortality of the Elves.
A few kindreds of Dwarves also fell under the Shadow, especially among those clans that lived in the mountains to the far east of Middle-earth. Some Dwarves fought on Sauron's side during the Last Alliance.
The term is also a euphemism that alludes to the seduction of Men by Morgoth in the beginning of their history: "Men came forth in the East of the world; but they fell under the shadow of the Enemy". This usage is similar to the other examples above, but is more far-reaching in that all Men living at that time were said to have succumbed to Morgoth's power.
In more general applications, "Shadow" is a term that apparently refers to the bewilderment and darkness accompanying certain enchantments. Sauron's tower of Barad-dûr was surrounded by Shadow, and the word is also associated with the land of Lórien and with the Ents.