Appearance and Characteristics
The Veil appears to be the plane of existence on Earth that ghosts, demons, hellhounds, and reapers are able to inhabit; and in the case of angels and demons occupying vessels, are able to pierce through and perceive the Veil, and can only be visible to those who can pierce/inhabit it.
Humans will begin to pierce through the Veil once a Crossroads Demon contract is up, allowing them to see hellhounds and the true faces of demons. Piercing the Veil can be incredibly taxing for a spirit and takes a lot of concentration which was evident from the struggle that Kevin went through to sustain his form on Earth. In addition to Heaven, Hell, and Purgatory rogue reapers appear to have secret ways in and out the Veil.
The ancient Celts believed that October 31st was the one night of the year where the Veil was the thinnest between the living and the dead. They used to appease the demon Samhain, who was able to summon ghosts (among other supernatural entities) from the Veil onto the earthly plane.
Kevin Tran revealed to Sam and Dean Winchester that after Metatron enacted his spell Heaven was locked down, forcing all spirits to remain stuck in the Veil at the sites of their deaths. Following Metatron's defeat, Heaven was restored and reapers now continue to deliver souls there. Despite this, Kevin himself remained stuck in the Veil for nearly two years before God sent his soul to Hell.
While creating a soul bomb, the Winchesters found that they lacked the necessary amount of souls to defeat The Darkness. Billie, however, was able to take about 200,000 souls from The Veil into the bomb. It is unknown what happened to those souls after they were absorbed by God from Dean.
There is a special part of the veil that appears as a library of books that each detail how someone dies. There is a particular section in the library that contains a multitude of different versions of Dean's death.
In Advanced Thanatology, its stated that ghosts have to cross the Veil in order to become visible. The ones that lack the strength to do so are the ones that cannot be seen.