Roots of a function f(x) are the values of x for which f(x) = 0.
To find the roots graphically we plot the graph f(x) versus x and find the points where the graph intersects the x-axis. The value of x at these points if substituted in f(x) gives 0.
Algebraically, roots can be found by solving the equation f(x) = 0.
For some forms like quadratic and cubic equations you have standard formulae for the roots. For higher order equations factorization has to be used. And they have to be reduced in order.
Example: For the function f(x) = x^2 - 2x - 3, the roots can be found by solving x^2 - 2x - 3 = 0
=> x^2 - 3x + x - 3 = 0
=> x(x - 3) + 1(x - 3) = 0
=> (x + 1)(x - 3) = 0
The roots we get are x = -1 and +3
If the graph y = x^2 - 2x - 3 were to be plotted we would find the graph intersecting the x-axis at x = -1 and x = 3.