Complexity of Newtons Method

Can anone here explain to me how to find the computational complexity of Newtons Method for roots?

...yeah ()()()))(())) ;)
 to be fair though, this bit was good:


For example, we can compute the square root of 2 as follows. Suppose our initial guess is 1:

Guess     Quotient     Average
 
1     (2/1) = 2     ((2 + 1)/2) = 1.5
 
1.5     (2/1.5) = 1.3333     ((1.3333 + 1.5)/2) = 1.4167
 
1.4167     (2/1.4167) = 1.4118     ((1.4167 + 1.4118)/2) = 1.4142
 
1.4142     ...    ...

Continuing this process, we obtain better and better approximations to the square root.

I would've liked more explanation on how/why this works for square roots, and how it was found though... rather than.. blah i'll shut up and google newtons method on wikipedia