Ever since Gilles Castel documented his method for taking math notes in real-time in Vim, I’ve noticed Emacs users asking if it’s possible to replicate this level of speed and convenience typing in Emacs1. Well, it’s Emacs. Of course all of this–and more–is possible. I write LaTeX all day, and I’m about as fast as when writing equations on paper. Sometimes a smidge faster. In this write-up I explain how, but first here’s what I mean:
Math input for the impatient:

Matrices in a hurry:

A typical statement:
The rest of this article outlines the components involved in LaTeX editing in Emacs and breaks down the features that are used to speed up text input specifically. The objective is to enable Emacs to let you type LaTeX, including math, as fast as you can type regular text or write math on paper. Fast enough to take live notes, at least.
You can use the TOC (to the right or below) to gain an overview of the tools involved. The demos might be helpful too.
The configuration code is available as a single file here. It will install three Emacs packages: AucTeX, CDLaTeX and YaSnippet. It does not install any YaS snippets. The snippets I use are here, but I recommend writing ones that suit your needs instead.
One of the features included in this code, used to rapidly input LaTeX tables or arrays, is available as a separate package here: Lazytab.
This is my rough understanding of LaTeX editing in Emacs:

Click to embiggen. Also every block is a link to the corresponding package.
At the core is the TeX document structure: the document tree, text and floats, the document class and style, cross-references and citations, other metadata and doc versions2. Surrounding it are the islands of concern when using a text editor to create this document: text input and text editing, document navigation, appearance, linting and compilation. The spokes leading into these islands are Emacs packages, libraries or library features that enable or hasten these tasks.
I’m not qualified to comment on most aspects of this map. Ask an Emacs user who’s written a book or two. This article is intended to be a collection of tips to speed up LaTeX editing to the point where you can take live notes, not a comprehensive guide to editing LaTeX in Emacs. Thus we’re mostly looking at the Text Input block, with quick dips into the Appearance and Text Editing sections. If you’re interested in the full picture I recommend starting with the AucTeX manual (C-h i m auctex in Emacs). It’s quite readable, and if you have the rare gene that lets you tolerate manuals, quite pleasant too!
On account of my ignorance, there are atleast a couple of big omissions in this chart. These are worth mentioning: