The roots of a polynomial are the places where the polynomial evaluates to 0 (zero). In other words, if one were to graph a polynomial, the roots would be all the places where the function crosses the x axis. For example, consider the function x + 1. To solve for the root of x + 1, set the whole function to 0:

0 = x + 1

and solve for x (x should evaluate to -1). With polynomials of greater degree, it gets a little more complicated. Here is x2 - 4, which is a polynomial of degree 2:

0 = x2 - 4

There is no easy way to solve this, but luckily, x2 - 4 factors into:

0 = (x+ 2)(x - 2)

and we find that the roots are 2 and -2 (plug each of these in for x and solve the equation and you'll see why). A polynomial of degree n has at most n roots in the real number system and always has exactly n roots in the complex number system. It has been proven that polynomials of a degree greater than 4 cannot be solved, but functions such as Newton's method can estimate roots to them.