The traditional legal institution behind the blood hunt is the Lextalionis (or Lex Talionis), a legal principle that can be summarized as "an eye for an eye". Such an order is usually given whenever a Kindred has committed a serious offense, such as kinslaying, Diablerie, breaching the Masquerade, invasion of domain that has resulted in any of the above offenses, or any other behavior that a prince deems to be a sufficient threat to the safety of the Masquerade and the Kindred of the city as a whole.
Any Kindred who hears the order for a blood hunt is expected to participate, although it's not required unless the Kindred in question has committed truly serious offenses (such as deliberately or repeatedly violating the Masquerade). Typically, the damned have until midnight to escape the city. If they don't, then they are open to retribution by their fellow Kindred. Any Kindred who provides assistance to the target of a blood hunt will often become the next to be hunted.
Some vampires use the blood hunt as a time to settle scores and get rid of rivals and enemies, and no few vampires have indulged in the chance to commit diablerie on their target or on other vampires who had the misfortune to get caught in the crossfire. Princes also have been known to use the Blood Hunt as a means to eliminate political or personal opposition as well, although this is, strictly speaking, forbidden, and the misuse of the blood hunt can have severe consequences on the prince in the long term.
A vampire who is placed on the Red List by the Inner Circle of the Camarilla is considered to have committed crimes so heinous that the entire sect wants the vampire's Final Death, and thus is considered to have a blood hunt on them in every domain of the Camarilla. Since the Anathema, as these vampires are called, are often vastly more powerful than the typical vampire or even a coterie of vampires, the Alastors are the ones who are charged by the Inner Circle with the task of hunting down and destroying these vampires.
The Sixth Tradition, the Tradition of Destruction, is used as the legal pretext for enacting the Lextalionis. Since the advent of the sects, Kindred are often subject to the blood hunt for aiding and abetting the Sabbat, or for subverting the Camarilla's hegemony in some other fashion. The Sabbat does not exactly practice the blood hunt, per se, as the sect disregards the Traditions in favor of the Code of Milan. When the Sword of Caine invokes the Lextalionis, it instead calls for a Wild hunt on the unfortunate target; the early Anarchs who preceded the Sabbat used the wild hunt in a similar fashion.
One of the greatest crimes a Sabbat can commit is to turn traitor, and the Sect protects its secrets. If a member reveals a Sect secret to the enemy, she is punished severely. If a Sabbat leaks information of a vital nature, a priest may call for a Wild Hunt. The Wild Hunt is much like the blood hunt, but ends with the eradication of the offending Sabbat Sect member, as well as anyone — kindred or kine — who may have knowledge of the betrayal. The gravity of the Wild Hunt depends upon the traitor in question — the packs are expected to police their own ranks, while high-profile turncoats receive the attention of Archbishops, Prisci, Cardinals, and all those who serve them.
The priest assembles the local Sabbat and formally calls them to the hunt, which is sometimes similar to the preface of a War Party. Once caught, the offending Sabbat is staked and immobilized. The pack takes her before the Ductus and Pack Priest (or Bishop, etc.), who recite her crimes to her packmates. The pack then torments the offender in whatever manner it deems appropriate — hot irons, Vicissitude, and mutilation are the least-creative forms of vengeance a righteous pack can inflict on a traitor. Finally, the pack destroys the traitor by throwing her (still staked) on a consecrated burning pyre.
After the traitor meets her Final Death, the Sabbat pursues those who either learned of the secret or were involved. Sabbat justice is relentless — the Sect stops at nothing to ensure their security. Naturally, the Sabbat cannot know about every little (or even many of the big) secrets that slip through the cracks. Frustration over this fact often makes things doubly bad for those they do catch.
Sabbat who are subject to the Wild Hunt are no longer Sabbat, and thus, no longer considered by the Sect to be vampires at all. No amount of groveling can convince the Sect to take back a traitor, though this harsh reality comes more out of security than bullheadedness.