Classical algebra was first developed by the ancient Babylonians, who had a system similar to our algebra. They were able to solve for unknown quantities (variables) and had formulas and equations. This may seem elementary, but many advanced civilizations solved such problems geometrically because it was more visual. This is similar to the idea of graphing two linear equations to see where they intersect rather than directly solving for the solution. The Chinese began to publish their own algebra writings around 100 BC.
Modern Algebra has come into existence much more recently, emerging over the past 200 years. This is a very complicated study of abstract ideas that are useful for mathematicians and scientists. It also includes some more basic topics like boolean algebra and matrix multiplication. Modern day physics and quantum physics rely heavily on the new concepts of modern, or abstract, algebra.
The word “Algebra” literally means the re-union of broken parts based on the origins of Arabic language. It was first used around 800AD by Arabic scholars, and is still in our language today. Once even the basics of algebra were only studied by advanced mathematicians and scientists in ancient civilizations, but now it is taught routinely to 7th and 8th graders.