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. Because Morgoth tainted Arda with his rebellious spirit, his will could continue to affect the hearts of people living on Earth after he was gone. Thus, societies that came under the sway of this corrupting power of Morgoth, or his servant Sauron, 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.