Algebra is the way we connect our ideas about geometry with our ideas about numbers. You can use equations to represent geometric shapes. You can also use equations to represent the movement of numbers (or real objects like trains) through time or space.
Equations can also have two variables. These look like y = 2x + 1 or x = 4y - 12. These equations also define a straight line, but the line can be tilted in various ways, and can move up or down, to the left or to the right.
Quadratic equations help you to define a curved line. They look like y = x2 or x = 8y3 + 16.These equations each define one particular line, but you can also use algebra to describe a whole set of lines that are all in the same category. Simple categories are called algebraic groups, and more complicated kinds of groups are called rings and fields.
To find out more about algebra, check out these books from Amazon.com or from your library: