The Blood Hunt is a Tradition among vampires of the Camarilla where the Prince of a city calls for the Final Death of another vampire. Only a Prince (or his superiors) can order a Blood Hunt, and it's usually reserved only for when a vampire has committed a serious offense, such as kinslaying, diablerie, breaching the Masquerade, aiding and abetting the Sabbat, subverting the Camarilla's hegemony, or any other behavior that the Prince considers to be a sufficient threat to maintaining the Masquerade and the safety of the vampire community.
Typically, the fugitive has until midnight to escape the city. If they don't, then they are open to retribution from their fellow vampires. Depending on the exact nature of the crime, the Prince may or may not authorize the diablerie of the fugitive. Even during a Blood Hunt, however, the Masquerade must still be preserved above all else.
Any vampire who hears of the Blood Hunt order is expected to participate, although it's not required unless the Kindred in question has committed very serious offenses (such as deliberately or repeatedly violating the Masquerade). Vampires who provide actual assistance to the fugitive, though, will often become the target of a Blood Hunt themselves.
Some vampires use the Blood Hunt as an opportunity to settle old scores and get rid of rivals and enemies, and more than a 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.
Although Princes have used the Blood Hunt as a means to eliminate political or personal enemies in the past, this is technically forbidden, and the misuse of the Blood Hunt can have severe consequences on the Prince in the long term.