The actual program that does the parsing. The common are pdfTex, XeTeX, LuaTex. They make use of a number of primitive instructions, like \def, \outer, \expandafter, \noexpand, \futurelet, \relax, \catcode, \vbox, \hbox, \accent, \kern.
Latex2e is a set of macros in use by over 20 years, most commmon. Compatible with Plain Tex. Code example:
\documentclass{article}
\begin{document}
Hello World.
\end{document}
Plain Tex is a set of macros used by D.Knuth to typeset his books. Compatible with Latex2e. Code Example:
Hello
\bye
ConTeXt is a specific format of LuaTex, incompatible with latex. Code Example:
\starttext
Hello World.
\stoptext
XeTeX is like latex2e, but with macroses for unicode. Basically, it requires \setromanfont{Ubuntu Mono} to show cyrillic.
Keywords are prepended by a backslash. According to Tex by topic, chapter 36.2 Keywords the list is: at, bp, by, cc, cm, dd, depth, em, ex, fil, height, in, l, minus, mm, mu, pc, plus, pt, scaled, sp, spread, to, true, width.
Chapter 37 is a glossary of primitives. Some of them might look like macros, but they're the primitives implemented in TeX engine.
Box numbers are limited, probably due to restrictions on PCs' power of the time of the specification. I think I could make them the size of native uint, and later put a silly question on tex.se about the limitations, mostly to promote kickstarter project.
\makebox (latex) means making a virtual box around the text. The idea is to arrange the text inside the box. For example: \makebox[30ex][s]{Censored text}\hspace{-30ex}\makebox[30ex][s]{X X X X X} here "Censored text" is arranged (spread) inside a box 30ex in width. Then the virtual caret moves back -30ex spaces, and the process repeated with text X X X X X. framebox does the same, but the box is visible (thickness is configurable though).
& at least in latex (?) is a mark to align a text to (it's easier to look up examples).
\usepackage is a latex macro. An analog in plain tex is \input, e.g.:
\input amstex %
\input harvmac %
Some text here
$$ y = mx + c $$
\bye
see https://tex.stackexchange.com/questions/71028/displaystyle-dfrac-dcases As I understand there're 4 styles 1. math block(D), 2. inline math(T), 3. subscript or superscript(S), 4. sub-subscript or super-superscript and further(SS).
States of the TeX parser automata, to differ {, $, usual characters, etc. More here: https://tex.stackexchange.com/questions/16410/what-are-category-codes