Vampires are creatures of magic, and tales in the Book of Nod tell of rituals and powers unknown to the common spate of Disciplines. For the most part such strange capabilities - controling the weather, altering one's own blood, casting mystical lights and so on - remained the province of apocryphal tales. A rare few vampires learned to harness the power of their blood to perform magic, but most used only the gifts granted by the Curse.
In the Dark ages, though, the Tremere clan rose to prominence through the use of Thaumaturgy. By combining their mortal wizardly knowledge with their new-found powers of blood, the Tremere discovered ways to control the manipulation of the vitae, forcing it to perform specialized powers. Furthermore, through careful rituals, they were able to create wide ranging effects, much like the powerful spells attributed to legendary sorcerers.
The Discipline of Thaumaturgy is the study of Vampiric magic, the ability to shape the forces of vitae into directed mystical energy. Pioneered primarily by the Tremere, the Discipline is actually composed of several "paths." Each path represents a particular area of magical study: blood manipulation, fire control, warding or other, stranger effects.
Thaumaturgy is the province of the Tremere clan, and they guard their wizardly secrets jealously. Indeed, learning even the rudiments of Thaumaturgy can take a year or more for vampires unskilled in the Tremeres arcane ways. The Tremere do not teach this Discipline to those outside their clan; vampires who wish to study Thaumaturgy must find a traitor to the Tremere or else engage in decades of intensive research with the rarest of occult materials and books. However, a few vampires outside the Tremere clan do study Thaumaturgy, Most notably, some Assamite visiers have learned to cast Thaumaturgy as a result of there studies of the Tremere curse and its underlying magic; however, as this magic is not native to them, their forms of Thaumaturgy always cost an extra Blood Trait. Within the Sabbat, some rare Tzimisce calling themselves Koldun study the ways of Thaumaturgy that were brought by the former Tremere defectors. Rumors hold that true Koldun actually study a complete different form of sorcerery, far older then Tremere Thaumaturgy, but none would ever wish to meet a Tzimisce old and twisted enough to know the truth of such matters.
In game terms, vampiric magic is usually referred to as Thaumaturgy, though not all Thaumaturgy is the same. A form of corrupting magis is supposedly practiced by the Followers of Set, for instance, yet it is totally different from and incompatible with the practices of Tremere magic. Of course, this does not stop the Tremere from using their knowledge to develop duplicates of such powers. This flexibility to devise paths (duplicating or opposing nearly any Discipline) is, perhaps, the greatest strength of Thaumaturgy.
Any use of a Thaumaturgical path requires the expenditure of a Blood Trait. Some specific powers may require additional expenditures as well. Thaumaturgy does rely on speaking magical phrases and gestures, and as such its use is generally noticeable.