The Veil is the gap between the mortal world an the next known as the Spirit world. The Spirit world is inhabited by the Dorocha and other creatures of the Old Religion, like the manticore (Love in the Time of Dragons) or the Fomorroh (A Servant of Two Masters). The Veil is guarded and kept by the Cailleach, goddess of death and winter.
On Samhain Eve, the Veil between the worlds is at its thinnest and the High Priestesses, back in the time of the Old Religion, would perform a blood sacrifice to tear the Veil and release the Dorocha. To heal the Veil, another sacrifice had to be made (The Darkest Hour).
It's possible that the Veil between the worlds were somehow involved in Morgause's ritual to summon Ygraine's spirit from the Underworld. Morgause chanted a spell that allowed her to stop time and Arthur was able to talk to his mother. The spell didn't hold long and the doorway (likely the Veil) was quickly closed, never to be opened again (probably only for Ygraine) (The Sins of the Father).
The Horn of Cathbhadh, a magical tool which had the power to summon the spirits of the dead, was used, long before the time of the Great Purge, each year at Beltane by the High Priestesses who would gather at the Great Stones of Nemeton to summon the spirits of their ancestors. In the days of the Old Religion, however, the priestesses trained for years before entering the Spirit World, since it was fraught with dangers. The one thing they were schooled never to do was to look back at the spirit, as the Veil closed. If they had done it, they would have released the spirit into this world.
Arthur Pendragon blew the Horn to talk to his father, entering an undefined region of unnatural bright light, but since he had looked back at him before returning to the world of the living, he had to use the magical tool again to reopen the Veil and force Uther into the Spirit world (The Death Song of Uther Pendragon).