Twilight

"Twilight is the absense of light. The Twilight Realm exists in a dimension where there is no light, hence, it exists only in Twilight.  Where there is no light, there is Twilight.  Where there was once light, but it is taken or lost, the light that once existed there fades away, and Twilight is the space that is left behind.  When an area loses it's light, Twilight is left behind and slowly grows as the light in the respective area diminishes."

-Great Sage Mojo-Jojo

Light in Hyrule is supported by the light carried by the Light Spirits, during the events in Twilight Princess, that light is taken away. And the province in which the light is stolen, slowly succumbs to the Twilight, as it's light fades away. Zant, the Twilight King, steals the Light form the Light spirits of Hyrule and Hyrule is slowly overtaken by Twilight as it's light vanishes.

Once overtaken by the Twilight the inhabitents of the Light are turned into lost spirits. They lose their physical bodies, and become spirits. They are for the most part, unaware of the events, but can see the shadow creatures and other actions performed by them. Once someone has entered the Twilight, it is often impossible to escape or recover. Only with the help of a member of the Twili race can lost soul or anything trapped in the Twilight escape. However, there are a select few that can survive in the Twilight without becoming a lost soul. Bearers of the Triforce can usually withstand the effects of the Twilight. Link, bearer of the Triforce of Courage, has been blessed by the Gods to survive the effects of the Twilight Realm. So instead of becoming a lost spirit, Link retains a physical form, however the form is that of a wolf.

Not all beings are turned into spirits however. Many are turned into 'shadow creatures' within Twilight and become mere shadows of their former selves, no hint of personality. For example, Bulbins that get caught within Twilight changed to twisted versions of themselves, with grey helmets and black/grey skin tones like what the Twili have, Bats, Deku Baba's and Kagaroks are just a few creatures that change in the Twilight Realm. Twilit creatures also have the ability to change people/spirits into monsters. As told in Twilight Princess, an old lady goes into the General Store in Kakariko Village and gets transformed into a shadow creature.

Twilight however, can be pushed back and eventually destroyed. This is done by returning the Light to the respective province by giving back the light to the Light Spirits. Once the light is returned to the light Spirits, light returns to the respective province and the Twilight vanishes.