Origins
Algebra is the branch of mathematics that generalises arithmetic by using symbols (letters, most commonly x, y, z for unknowns and a, b, c for constants) to represent numbers and quantities, and that studies the rules governing the manipulation of those symbols. In its modern abstract form, algebra has expanded far beyond the solving of equations to investigate whole structures — groups, rings, fields, modules, vector spaces, algebras over fields — of which ordinary numbers are only one example. The word comes from Arabic al-jabr (الجبر), "the reunion of broken parts, the restoration," the title-keyword of the ninth-century treatise by Muḥammad ibn Mūsā al-Khwārizmī that laid the foundations of the discipline. Through the Latinisation of al-Khwārizmī's own name the same treatise also gave Europe the word algorithm, so that the two great loanwords of medieval Islamic mathematics descend from a single book.
The key text is al-Khwārizmī's al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wa l-Muqābala ("The Compendious Book on Calculation by Restoration and Balancing"), written in Baghdad around 820 CE at the Bayt al-Ḥikma (House of Wisdom) under the Abbasid caliph al-Maʾmūn. The title names two complementary operations: al-jabr, the "restoration" by which negative terms are moved across the equals sign and made positive, and al-muqābala, the "balancing" or "confrontation" by which like terms on opposite sides are cancelled. Al-Khwārizmī's treatise presented a systematic classification of linear and quadratic equations into six canonical forms and gave general methods of solution, and it was built on earlier work by Diophantus of Alexandria in the third century, Brahmagupta in seventh-century India, and on the geometrical algebra implicit in Euclid. But al-Khwārizmī was the first to treat algebra as an independent discipline rather than as an adjunct to geometry or arithmetic, and this step is what makes him, in the European tradition, the father of algebra.
The word al-jabr itself had a medical meaning before it had a mathematical one. Classical Arabic jabara means "to set a broken bone," "to restore something fractured," and more generally "to compel, to force, to make whole by compulsion." An al-jabbār (from the same root) was both a bone-setter and, in theological contexts, an epithet of God ("the Compeller, the Restorer," one of the ninety-nine names). Al-Khwārizmī's use of al-jabr for the algebraic operation of moving a term across the equation was therefore a medical metaphor: a broken equation is set back together. This origin is preserved, curiously, in medieval and early-modern Spanish, where algebrista continued to mean both "mathematician" and "bone-setter" well into the seventeenth century. Don Quixote (part II, 1615) speaks of "un algebrista que curó al pobre Sansón" — an algebrista who healed poor Sansón Carrasco — meaning a bone-setter, not a mathematician. Barber-surgeons' shops in Spain into the nineteenth century sometimes advertised algebrista among their services.
Latin Roots
The word entered Latin Europe through two principal translators. Robert of Chester translated al-Khwārizmī's treatise into Latin in 1145 as Liber algebrae et almucabola; Gerard of Cremona made a second Latin version from Toledo in the later twelfth century. These translations circulated in Italian and Iberian university circles through the thirteenth and fourteenth centuries. Italian abacus schools — the scuole d'abbaco of Florence, Pisa, and other mercantile cities — absorbed algebraic method into the practical mathematics of commerce, and by the early sixteenth century Italian mathematicians were pushing beyond al-Khwārizmī: Scipione del Ferro and Niccolò Tartaglia solved the general cubic (c. 1515–1535), Ludovico Ferrari the general quartic (1540), and Gerolamo Cardano's Ars Magna (1545) published these results. The word algebra is first attested in English in Robert Recorde's The Whetstone of Witte (1557) — the same book that introduced the equals sign — where Recorde calls it "the rule of Coss," glossing the Italian cosa (the unknown, "the thing"). Shakespeare refers to algebra in passing in The Merchant of Venice; Samuel Johnson's Dictionary (1755) defines algebra tersely as "a peculiar kind of arithmetick, which takes the quantity sought, whether it be a number or a line, or any other quantity, as if it were granted."
Modern algebra diverges from its medieval ancestor in two major directions. Elementary algebra is the school subject, concerned with symbol manipulation, linear and quadratic equations, polynomials, and systems. Abstract algebra, developed from the nineteenth century onward by Évariste Galois (whose work on solvability of equations gave the first group theory in the 1820s–1830s), Arthur Cayley, Emmy Noether, and others, studies algebraic structures as objects in their own right: groups, rings, fields, modules, lattices, Lie algebras, Hopf algebras. Boolean algebra (George Boole, 1854) generalises algebra to propositions. Linear algebra (vector spaces and linear maps) is now a foundational component of engineering, physics, machine learning, and statistics. Computer algebra systems (Macsyma, Mathematica, Maple, SageMath) have automated much of the symbolic manipulation that defined the subject from al-Khwārizmī to Newton. The word itself, still audibly Arabic after twelve centuries, remains one of the clearest marks in European languages of the debt that Latin science owes to the Islamic mathematical tradition.