The word "algebra" comes from a phrase (in bold below) in the title of an Arab book "Kitab al muhtasar fi hisab al gabr w'al muqubalah." This has been translated as "A compact introduction (book) to calculation using rules of completion and reduction," but Solomon Gandz has suggested "al gabr" comes from Babylonian "gabru" meaning solution of an equation, and that "muqubalah" (q reads like k) was its equivalent in Arabic. The book covered simple equations like the one in the preceding section, also quadratic ones involving x2, as well as other areas such as geometry and the division of inheritances.
Its author, Mukhammad ibn Musa Al-Khorezmi (lived about 780-850) was the chief mathematician in the "House of Wisdom", an academy of sciences established in Baghdad by the Caliph Al Ma'mun, son of Harun Al Rashid of "Arabian Nights" fame. The "House of Wisdom" was involved in Al Ma'mun's expedition to measure the size of the Earth, which Al-Khorezmi afterwards estimated to have a circumference of 21000 Arab miles. (We are not sure how big the Arab mile was, the actual figure is about 25000 of our miles; more about such estimates, here).
Al-Khorezmi's family (and possibly he as well) apparently came from the oasis of Khorazem, at the southern end of the Aral Sea, in what is now Uzbekistan. He is also credited with helping establish among the Arabs the Indian numbering system, using decimal notation and the zero. Previous systems of writing numbers used letters, like the Roman numeral systems or the cruder ones of the Greeks and Hebrews. When Al-Khorezmi's book on the new system reached Europe, the Europeans called its use "algorism" or "algorithm," a corruption of the author's name. Today "algorithm" means method of calculation, and the rise of computers has led to extensive work on developing efficient computer algorithms.
More about Al-Khorezmi's work, here.