I was watching a fascinating video about the origins of algebra on the Khan Academy website. I’d recommend this website to anyone who wants to bone up on math or the sciences (either for school or for your own information).
Turns out, algebra comes from “al jabr” which is the Persian/Arabic word meaning restoration or completion. A man named Muhammad ibn Mūsā al-Khwārizmī (c. 780–850) wrote the first algebra book “The Compendious Book on Calculation by Completion and Balancing”, which established algebra as a mathematical discipline that is independent of geometry and arithmetic.
Source cited: http://en.wikipedia.org/wiki/Algebra#Rings_and_fields