No one actually knows where these immortals come from. They are found as children, adopted by mortals and grow up quite normally, though immune to disease or infection, until their first death. After that they wake again and no longer age. Wounds heal quickly and without scarring, except for wounds at the neck/throat. They can only be killed if their heads are cut off.
The Game refers to the process of immortals fighting elimination duels until only one is left standing.
The Gathering refers to the effect that immortals feel driven to search out other immortals and challenge them.
A Quickening is what happens when an immortal's head is cut off by another. That immortals memories and strength are transferred to the winner.
A Dark Quickening is what happens when the killed immortal's personality overwhelms the winning immortal. This may happen both ways: Turning an evil person good or vice versa.
Holy Ground, no matter the religion/belief it belongs to, is safe. Immortals do not fight on Holy Ground. It has been tried, and the effects of a Quickening there are devastating. See Pompeii.