Algebra (from Arabic al-jebr meaning "reunion of broken parts") is one of the broad parts of mathematics, together with number theory, geometry and analysis. 

As such, it includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. 

Algebraic expression notation: 1 – power (exponent) 2 – coefficient  3 – term 4 – operator  5 – constant term x y c – variables/constants. image: wikipedia

The more basic parts of algebra are called elementary algebra, the more abstract parts are called abstract algebra or modern algebra. Elementary algebra is essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. 

Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians. Much early work in algebra, as the origin of its name suggests, was done in the Near East, by such mathematicians as Omar Khayyam (1050-1123).

Difference between algebra and arithmetic 

Elementary algebra differs from arithmetic in the use of abstractions, such as using letters to stand for numbers that are either unknown or allowed to take on many values. 

For example, in x + 2 = 5 the letter x is unknown, but the law of inverses can be used to discover its value: x=3. 

In E=mc^2, the letters E and m are variables, and the letter c is a constant. 

Algebra gives methods for solving equations and expressing formulas that are much easier (for those who know how to use them) than the older method of writing everything out in words.

Other ways the word 'algebra' is used

The word algebra is also used in certain specialized ways. A special kind of mathematical object in abstract algebra is called an "algebra", and the word is used, for example, in the phrases linear algebra and algebraic topology .

A mathematician who does research in algebra is called an algebraist.