The rate of convergence is an attempt to quantify the answer to the question How quickly does the algorithm zoom in on the root? Formally, look at
where the constant 
is the convergence constant and the
exponent p is the convergence rate. For example, linear convergence
says the error is reduced by a fixed amount with each iteration -
say cut in half each iteration.
Quadratic convergence says the error at the kth iteration is the square
of the error at the (k-1)-st iteration.
The convergence rate is important if you want to estimate
how many iterations it will take to reach a given relative error.
Although the arithmetic is messy, you should show that, to obtain
a relative error of 
, linear convergence with a convergence
constant of 0.8 would take about 30-32 iterations, and cubic convergence would
take about 3 iterations. How many would quadratic convergence take?