Algebra is (according to the Oxford English Dictionary, OED) 'the department of mathematics which investigates the relations and properties of numbers by means of general symbols; and, in a more abstract sense, a calculus of symbols combining according to certain defined laws'. Algebra was started by al-Khwarizmi and involves using letters, called variables, to represent numbers that change, and modern algebra now uses a grid graph, and together these ideas are used to study functions, which have many scientific applications.
More advanced types of algebra includes linear algebra, which is (according to OED) algebra 'in which the symbols are linearly connected,' algebra of logic, which is (according to OED) algebra 'in which the symbols represent not numbers or quantities, but other objects of thought, as classes or qualities of things, or statements concerning things,' and abstract algebra.
To do difficult but very practical maths, or to be a mathematician, one should study algebra (after basic arithmetic) because all advanced, including some applied (such as in science) maths depends on it.
Two important algebraic equations are below.
slope-intercept form: y=mx+b
point-slope form: y-y1=m(x-x1)
An important algebraic expression is below.
polynomial of degree n and coefficients an to a0: anxn+...+a2x2+a1x+a0 (for degree two, for example, it would be more easily written ax2+bx+c)