Bharata Krsna Tirtha, a Sankaracarya (head of a Hindu monastery) was distressed that traditional mathematical computations were so cumbersome. After contemplation of the Vedas (the religious texts of ancient India), he was inspired to develop easier techniques for doing mathematics.
BKT summarized his discoveries into sixteen "sutras" (pithy aphorisms) such as "all from 9 and the last from 10" and "vertically and crosswise". Vedic Mathematics enables the student to perform complex arithmetical computations, often getting the answer in just one line.
Vedic Mathematics has been the subject of some criticism: some of BKT's shortcuts are a repackaging of existing techniques. (For example, his method of solving simultaneous linear equations is basically Cramer's Rule.) Others apply to such a narrow range of situations that they hardly merit a separate theorem.
Moreover, while BKT did give a few proofs of his results, on the whole he focused on the "how" rather than the "why" of his techniques - explaining why Vedic Mathematics has yet to gain widespread acceptance outside of India.
But, when Vedic Mathematics is good, it's great. There is a method of long division which - unlike our current system - doesn't depend on trial-and-error. There is also a remarkable notation to indicate deficiencies (for example, to express a number like 38 as "2 less than 40").
"Vedic Mathematics" should be required reading for all aspiring mathematics teachers.