Stillwell gives some details in Mathematics and its History. In modern terms, Newton made a general transformation of axes reducing the general cubic to four equation types, and then classified them into "species" according to the roots of the polynomials on the right hand side.
"His paper does not contain detailed proofs; these were supplied by Stirling (1717), along with four of the species Newton had missed. Newton’s classification was criticized by some later mathematicians, such as Euler, for lacking a general principle. A unifying principle was certainly desirable, to reduce the complexity of the classification. And such a principle was already implicit in one of Newton’s passing remarks, Section 29, “On the Genesis of Curves by Shadows.”"
The principle Stillwell has in mind is the projective classification over the complex field, but that is "implicit" indeed, no wonder Euler missed it.
Guicciardini in Isaac Newton on Mathematical Certainty and Method remarks that "the reader is not informed on the methods Newton deployed" and gives examples of Newton's claims "with no trace of demonstration", which Stirling's 1717 commentary fills in. I suppose this explains the gaps. Cayley gives a more detailed and less hindsighted description, as well as comparison to Plücker, in his On Classification of Cubic Curves (1864) reproduced in volume 5 of his Collected Mathematical Papers:
"The classification is according to the nature of the infinite branches; there are fourteen genera containing together seventy-two species, but four species were added by Stirling in his Linere Tertii Ordinis Newtoniare; sive Illustratio &c (1717), and two more by Murdoch or Cramer [Cayley's footnote: The two additional species are, I believe, first mentioned in Murdoch's Genesis Curvarum per Umbras (1746), but one of them is there ascribed to Cramer], making in all seventy-eight species. A new classification was made by Plücker in his System der Analytischen Geometrie, 1835; this is likewise according to the nature of the infinite branches, but after his six head divisions, and some subordinate divisions thereof, Plücker establishes the divisions called Groups, which have nothing analogous to them in the Newtonian theory; there are sixty-one groups, and the total number of species is 219."
If you really want to delve into it there is a very detailed commentary by Rouse Ball in On Newton's Classification of Cubic Curves (1890), who details how "four canonical forms or cases are divided into seven (or six) classes. These classes are subdivided into fourteen (or thirteen) genera, which contain the seventy-eight species, from which however the degenerate forms of a conic and a straight line and of three straight lines are excluded", and also tells us that
"Cramer, in his Introduction a l'Analyse des Lignes Courbes Algebriques, published at Geneva in 1750, and Euler, in his Analysis Infinitorum, published at Lausanne in 1748, proposed to classify curves solely by reference to the character of their infinitely distant points. They both refused to recognise the various curves formed by the degeneration of an oval into an acnode, or its coalescence with other branches of the curve, as constituting distinct species."
