Puiseux series and Resolution of Singularities

Question

I have a very basic knowledge of algebraic geometry(no schemes!), and am trying to study the resolution of singularities. So the Newton's method gives us a Puiseux series parametrizing the branches of a curve at a point. What I do not understand is how this gives a resolution of singularities. Since the Puiseux series is a power series, where is the variety here? I know this is a basic question, but most of the references skim through this aspect and make it sound self-evident. I am hoping for some simple explanation of the connection. Thanks!