The Dark Gift is a term for the vampiric power. When a vampire maker bestows the power to become immortal that is transmitted through the Blood on a fledgling, that maker is offering the Dark Gift. This is also sometimes referred to as The Dark Trick.
According to the rules of Santino's coven, the Dark Gift was to be given only to beautiful mortals, as an insult to God. Although most of the vampires in The Vampire Chronicles do indeed select beautiful victims, they are motivated by love rather than any rule.
The way the Dark Gift manifests varies, as it all depends on the siring vampire's age, power and blood purity. If a vampire has gone centuries without turning someone, then the vampire they turn will be incredibly powerful. Not all progeny display the same gifts: Louis for example, lacked telepathy.