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.
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. Magnus alone is old and ugly, which is why he had stolen the Dark Gift, for he had never been targeted for immortality.
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 either. Louis for example, lacked telepathy.
The numerous gifts that vampires can develop usually come with age though a vampire can take a shortcut to such power by drinking the blood of ancient and powerful vampires. Lestat became far more powerful and developed abilities sooner than he normal would after drinking the blood of Akasha.