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Why is it called Algebra?

In the 7th and 8th centuries the Arabs, united by Mohammed, conquered the land from India, across northern Africa, to Spain. In the following centuries (through the 14th) they pursued the arts and sciences and were responsible for most of the scientific advances made in the west. Although the language was Arabic many of the scholars were Greeks, Christians, Persians, or Jews. Their most valuable contribution was the preservation of Greek learning through the middle ages, and it is through their translations that much of what we know today about the Greeks became available. In addition they made original contributions of their own.

They took over and improved the Hindu number symbols and the idea of positional notation. These numerals (the Hindu-Arabic system of numeration) and the algorithms for operating with them were transmitted to Europe around 1200 and are in use throughout the world today.
Like the Hindus, the Arabs worked freely with irrationals. However they took a backward step in rejecting negative numbers in spite of having learned of them from the Hindus.
In algebra the Arabs contributed first of all the name. The word "algebra" come from the title of a text book in the subject, Hisab al-jabr w'al muqabala, written about 830 by the astronomer/mathematician Mohammed ibn-Musa al-Khwarizmi. This title is sometimes translated as "Restoring and Simplification" or as "Transposition and Cancellation." Our word "algorithm" in a corruption of al-Khowarizmi's name.

Learn more about one of the Fathers of Mathematics here:

http://www-history.mcs.st-and.ac.uk/Biographies/Al-Khwarizmi.html