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TeX, pdfTeX, XeTeX, LuaTeX, LaTeX, pdfLaTeX, XeLaTeX, LuaLaTeX Explained

This article is a TL;DR version of What's in a Name: A Guide to the Many Flavours of TeX if you want to find a quick understanding of them. I highly recommend reading the original article for more details.

TeX is the original typesetting system (program/engine that handles typesetting problems, including: automatic line breaking, hyphenation and mathematical typesetting) written by Donald Knuth that dates back to 1970s. Currently, it is under maintenance mode and does not accept new features (only bug fixes, a decision made by Knuth in the 1980s). The output of TeX is DVI (DeVice Independent format).

Softwares derived from TeX to support new features (new output formats, font types and Unicode) receive names by adding a prefix to the word "TeX", such as:

These three are all TeX engines.

LaTeX is not another derivation from TeX (i.e., not an typeset engine like pdfTeX), it is a large collection of so-called TeX macros (combining engine's primitives or existing macros). It is designed to be extensible: you can plug-in additional, specialist, macro packages. LaTeX is written by Leslie Lamport in the mid-1980s. LaTeX macro package is still actively developed like TeX engines.

Typesetting with LaTeX means using LaTeX macro package with a particular TeX engine:

These are not TeX engines, all they signify is which TeX engine is being used to run the LaTeX macro collection.

Installation

TeX Live is the TeX distribution maintained by members of the TeX community, which is released yearly. It contains all the above mentioned TeX engines with other TeX-related tools, fonts and specialist LaTeX packages.

TeX Live provides commands: