Newton and Newton polygon What did Newton himself do, so that the "Newton polygon" method is named after him?
 A: 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. 88-104) and Examples (pp. 104-112). 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.
http://books.google.com/books?id=6I97byFB3v0C
http://name.umdl.umich.edu/ABQ9451.0001.001
A: The Newton polygon and Newton's method are closely related. The following theorem was first proven by Puiseux:

if $K$ is an algebraically closed field of characteristic zero, then the field of Puiseux series over $K$ is the algebraic closure of the field of formal Laurent series over $K$

However according to Wikipedia

This property was implicit in Newton's use of the Newton polygon as early as 1671 and therefore known either as Puiseux's theorem or as the Newton–Puiseux theorem.

A place where this is illustrated in more detail is "A history of algorithms: from the pebble to the microchip" By Jean-Luc Chabert, Évelyne Barbin, page 191. I will quote the first paragraph

Immediately following his description of his numerical method for solving equations, Newton used the same principle to show how to obtain algebraic solutions of equations. He explains how the method of successive linear approximations can be adapted by using a ruler and "small parallelograms", the first version of what is called Newton's polygon. The method was applied in a more general case later, by Puiseux in 1850, both in considering multiple branches and in considering functions of a complex variable

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.
A: here are some references:


*

*"Plane Algebraic Curves" by Brieskorn and Knorrer, around page 370.

*"Plane Algebraic Curves" by Fischer, Appendix 4. 

*The link 
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.
(However, it is a monster of a book.) 
A: 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.

Historical Note. - As has already been remarked, the fundamental idea of
  the  reversion of series, and of the
  expansion of the roots of algebraical
  or other  equations in power-series
  originated with Newton. His famous"
  Parallelogram" is  first mentioned in
  the second letter to Oldenburg; but is
  more fully explained  in the
  Geometria Analytica (see Horsley's edition of Newton's Works, t. i.,  p.
  398). The method was well understood
  by Newton's followers, Stirling and 
  Taylor; but seems to have been lost
  sight of in England after their time.
  It was  much used (in a modified form
  of De Gua's) by Cramer in his
  well-known Analyse  dea Lignes
  Courbea Algebriques (1750). Lagrange
  gave a complete analytical form  to
  Newton's method in his "Memoire sur
  l'Usage des Fractions Continues,"
  Nouv.  Mem. d. l'Ac. roy. d. Sciences d. Berlin (1776). (See OEuvres de
  Lagrange, t. iv.) 
Notwithstanding its great utility, the
  method was everywhere all but
  forgotten  in the early part of this
  century, as has been pointed out by De
  Morgan in an  interesting account of
  it given in the Cambridge
  Philosophical Transactions,  vol.ix.
  (1855). 
The idea of demonstrating, a priori,
  the possibility of expansions such as
  the  reversion-formulae of S.18
  originated with Cauchy; and to him, in
  effect, are due  the methods employed
  in SS.18 and 19. See his memoirs on
  the Integration of  Partial
  Differential Equations, on the
  Calculus of Limits, and on the Nature
  and  Properties of the Roots of an
  Equation which contains a Variable
  Parameter, 
  Exercices d'Analyse et de Physique Mathematique, t. i. (1840), p. 327;
  t. ii.  (1841), pp. 41, 109. The form
  of the demonstrations given in SS. 18,
  19 has  been borrowed partly from
  Thomae, El. Theorie der Analytischen
  Functionen  einer Complexen
  Veranderlichen (Halle, 1880), p. 107;
  partly from Stolz, 
  Allgemeine Arithmetik, I. Th. (Leipzig, 1885), p. 296. 
The Parallelogram of Newton was used
  for the theoretical purpose of
  establishing the expansibility of the
  branches of an algebraic function by
  Puiseaux in  his Classical Memoir on
  the Algebraic Functions (Liouv. Math.
  Jour., 1850).  Puiseaux and Briot and
  Bouquet (Theorie des Fonctions
  Elliptiques (1875), p. 19)  use
  Cauchy's Theorem regarding the number
  of the roots of an algebraic equation 
  in a given contour; and thus infer the
  continuity of the roots. The
  demonstration  given in S.21 depends
  upon the proof, a priori, of the
  possibility of an  expansion in a
  power-series; and in this respect
  follows the original idea of  Newton. 
The reader who desires to pursue the
  subject further may consult Durege, 
  Elemente der Theorie der Functionen einer Complexen Veranderlichen
  Grosse,  for a good introduction to
  this great branch of modern
  function-theory. 
The applications are very numerous,
  for example, to the finding of
  curvatures  and curves of closest
  contact, and to curve-tracing
  generally. A number of  beautiful
  examples will be found in that
  much-to.be-recommended text-book, 
  Frost's Curve Tracing. -- G. Chrystal: Algebra, Part II, p.370

