Around 820 CE, the Persian-Arab mathematician Muhammad ibn Musa al-Khwārizmī, working in the Baghdad House of Wisdom, wrote a textbook called Kitāb al-mukhtaṣar fī ḥisāb al-jabr wa-l-muqābalah — "The Compendious Book on Calculation by al-jabr and Balancing." It is the founding work of modern algebra, and the book that gave the discipline its name.
What al-jabr Meant
The Arabic al-jabr means literally "the reuniting of broken parts" or "the restoration." Al-Khwārizmī used it for the operation of moving a negative quantity to the other side of an equation, where it becomes positive — effectively "restoring" what had been subtracted. Together with al-muqābalah ("balancing"), it described two basic moves for solving an equation. Centuries later, the second operation was forgotten and only "al-jabr" stuck.
A Surgical Side-Meaning
One curious detail: in classical Arabic, al-jabr also referred to the medical practice of setting broken bones. The metaphor is the same — restoration of broken parts — and in medieval Spanish, the word algebrista meant both an algebraist and a bone-setter. Spanish barber-surgeons sometimes hung "Algebrista" signs outside their shops well into the 17th century.
Into Latin and English
Al-Khwārizmī's book was translated into Latin in the 12th century, in the wave of translations that followed the Christian re-conquest of parts of Spain. The Latin translators kept the Arabic al-jabr as algebra, and the discipline kept the name. Al-Khwārizmī's own name, Latinised as Algorismi, gave us the English word algorithm. So a single 9th-century Persian mathematician is the unwitting source of two of the most important words in modern computing and education.