Above the Human world
The Shinigami Realm is where Shinigami typically live and dwell throughout their existence. The Shinigami Realm has a connection to the Human World, which Shinigami are able to travel to and remain in for as long as their Death Note is in a humans possession.
The world that Shinigami reside in exists above the Human World. For example, Ryuk has to travel underground in order to reach the entrance to the realm of humans underneath. Also, when Ryuk is first seen after leaving the Shinigami Realm, he is flying downward from a higher altitude than the skyscrapers.
Appearance and lifeforms
The realm is dark and appears lifeless. Very little, if any, vegetation is shown in this barren land. Plants are never shown in the realm, though they exist as seen when Ryuk brings an apple to the Human World, although it was very dried up. Sentient beings other than the Shinigami do not seem to exist there.
The realm includes different landscapes and does not have human-like buildings. Shinigami tend to congregate in areas with bone and/or stone structures. There are also several areas with chains, most noticeably the Shinigami King's cave, where there are hundreds of chains with hooks on the end. There is at least one mountain or cave structure where Ryuk is living, which is shown in the one-shot special of the anime. This structure could also possibly be where the Shinigami King resides.
The realm is surrounded by deserts that have various strange objects. The spheres, which are shaped like concentric orbs, are littered about the surface. These artifacts have openings which let Shinigami see anywhere in the Human World from a birds-eye view. These strange objects are what allow Shinigami to take human lives with Death Notes without traveling to Earth. When or how the objects were created is currently unknown.
It appears that there is not much to do in the Shinigami Realm. Shinigami usually gamble with one another, watch the events occurring in the Human World through the spheres, terminate human lives with their Death Notes in order to prolong their own or eat (although they don't need to). This is also Ryuk's primary reason for dropping his Death Note into the Human World.