No single person discovered algebra, since various people in different parts of the world discovered it at different times. Some aspects of algebra were even discovered multiple times by different people who were unaware of each other. Virtually every major civilization worked out some portion of the algebraic puzzle, although certain people like Diophantus, Muhammad Ibn Musa al-Khwarizmi and Gottfried Leibniz made more significant contributions.
The Babylonians pioneered many of the basic usages of algebra. A tablet dated between 1900 and 1600 B.C. contains Pythagorean triples and other advanced mathematics. There is also evidence of rudimentary algebra in Ancient Egypt, including a document on linear equations that is one of the earliest mathematical proofs ever discovered. While the Ancient Greeks were better known for other forms of mathematics, they did devise a form of geometric algebra that used the sides of objects to represent algebraic terms. Mathematicians from present day India and China also developed early versions of algebra, with the modern algebraic term "Modus Indorum" referring specifically to an algebraic method devised in India.
Two of the most important people in the history of algebra are Diophantus and al-Khwarizmi. The former is frequently referred to as "the father of algebra," and his treatise "Arithmetica" was the first to use symbols to represent unknown numbers. Al-Khwarizmi was the first to distinguish algebra from geometry and arithmetic, and he pioneered the concept of balancing and reducing the sides of an equation. The word algebra refers to his work "Hidab al-Jabr wal-Muqubala," or "The Book of Restoration and Balance."
Since then, Europeans like Francois Viete and Gottfried Leibniz and scholars from other parts of the world, such as Seki Kowa, have refined mathematicians' understanding of algebra.