What did Newton himself do, so that the "Newton polygon" method is named after him?

The Newton polygon and Newton's method are closely related. The following theorem was first proven by Puiseux:
However according to Wikipedia
A place where this is illustrated in more detail is "A history of algorithms: from the pebble to the microchip" By JeanLuc Chabert, Évelyne Barbin, page 191. I will quote the first paragraph
Then it explains Newton's approach in detail. You can follow the references given there. And then we know the story that this nice tool is now used for the understanding of polynomials over local fields even tough originally the local field was the field of formal Laurent series. 


here are some references:
https://www.maa.org/sites/default/files/pdf/upload_library/22/Polya/07468342.di020774.02p0260v.pdf is an article from The College Mathematics Journal about the Newton polygon as developed by Newton. I especially recommend the first reference because it has a wealth of pictures. 


If memory serves correct the history of Newton's polygon and Puiseaux series has some subtleties, so be a bit wary of secondary historical sources. Histories of mathematics are bursting at the seams with romanticized legends, so it is always best to consult primary sources if you wish to know the real history. The following note from Chrystal's Algebra may serve as a helpful entry into the primary literature.



This was intended to be a comment on Bill Dubuque's answer, but I apparently don't yet have enough reputation points to comment, and in any event this is probably too long to appear as a comment. Given Chrystal's intended audience, I'm surprised that he didn't mention Talbot's 1860 English translation and extensive commentary of Newton's Enumeration Linearum Tertii Ordinis. In Talbot's work, which is freely available on the internet, see the sections On the Analytical Parallelogram (pp. 88104) and Examples (pp. 104112). By the way, whoever scanned the book for google wasn't paying attention when the lengthy list of figures at the end of the book were being scanned, so I'm also giving the University of Michigan Historical Math Collection version, which has those figures correctly scanned. Sir Isaac Newton's Enumeration of Lines of the Third Order, Generation of Curves by Shadows, Organic Description of Curves, and Construction of Equations by Curves, Translated from the Latin, with notes and examples, by C.R.M. Talbot, 1860. 

