What is LaTeX? and TeX?
TeX is a typesetting language that is designed to facilitate typesetting mathematical expressions. Many mathematics and physics journals and book publishers use TeX to set their pages. LaTeX is essentially a collection of macros that provide a simpler interface for using TeX. Although originally intended to be used for mathematics, TeX and LaTeX have many typesetting features that make it useful even if you aren't including any mathematics in your document. These include embedding graphics, tracking cross-references (for figures, tables, sections, subsections, citations), automatically generating reference lists in any desired bibliographical format (using BibTeX), creating simple line-art, producing globally consistent and well-structured format, etc. It is, in a single word, slick.
There are many reasons to use LaTeX and even some good reasons not to. LaTeX is without question the best option available for generating math-heavy documents. I have used LaTeX to write papers, create slide presentations, make quizzes/exams/homework sheets, design posters, produce letters with self-contained letterhead (example .tex, .pdf), assemble CVs etc. I have found it useful for all of these but no longer use it for slide presentations or posters as I have found more convenient options for these, Keynote and Illustrator respectively, although LaTeX Equation Editor does come in handy for including math in the former. If you want to read some entertaining and often dogmatic polemic on LaTeX versus Word, try Googling that phrase.
Overall procedure for using LaTeX
Generating documents with LaTeX requires three things. First, you have to produce a .tex file with the desired document content and formating commands. Some examples and links to online help with LaTeX commands are provided below. The .tex file must be a simple ASCII file so no matter what text editor you use, be sure to save it as plain text.
Next, you need to install a TeX distribution. This is a large set of files that contain all the code for turning your .tex file into a .pdf file. Links for help with installing can be found below. In some cases, this can be done with minimal button-pushing and near-complete ignorance on the part of the user so don't start sweating yet. Once the distribution is installed, provided you use a GUI LaTeX editor, you will not need to directly access the distribution and it would be best to leave it exactly where the installation procedure put it.
Finally, you need to typeset your .tex file. This can be done using the command line on most systems but that is so passé. Many GUIs exist for typesetting and most of them come in the form of a text editor so you can generate and typeset your file all in one place. Some of these are converging on a WYSIWYG environment. TeXShop (link below), available for the Mac, is what I use. I have posted links below to a few similar programs for Windows and Linux that have been suggested by others. Several people have recommended LyX (link below) describing it as a good intro for those new to LaTeX.
How to install TeX and typesetting GUIs
Being a Mac user, I have not used some of the (non-Mac relevant) pages linked to below but they look like they explain LaTeX and the installing procedure well enough.
"How to" pages
Getting a TeX distribution
Getting a LaTeX editor
Sample .tex files
The best way to learn LaTeX is to start with a sample .tex file and modify it to suit your needs. The following links are to .tex files that have some common techniques. You will probably want to look at both the .tex file and the resulting .pdf to understand what's been done.
How to use LaTeX
Once you have managed to modify one of the sample files to suit your needs, you might want to learn more about the commands you're using or find new ones. Google is a surprisingly useful way to do this; just be sure to include "latex" in your search string (e.g. 'umlaut latex'). These links might also help.
How to use BiBTeX
BiBTex is a tool for automatically generating bibliographies and correctly setting up the cross-referencing to your reference list in your document. This is one of the most powerful time-saving features associated with LaTeX. Here is a .tex and .pdf file that explain how to do it.
Other LaTeX tools
These "typesetting" programs can be used to generate mini-documents that can be dropped into a Keynote or Powerpoint presentation. I find LaTeX Equation Editor a nice way to send complicated math expressions by email (drag and droppable!).
If you find any online resources that are better than ones listed here or include material that is not covered here, please send me an email so I can update this page. Comments on what you found more/less useful on this page are also welcome as I would like to streamline this page for the LaTeX novice.
Links to add:
Sections to add:
Common errors (math mode vs text mode, nesting, delete aux files, file names with spaces)
List of useful .tex files and commands:
Matrices and vectors
Quotation marks in LaTeX - use `` or ` for opening quotations and '', ' or " for closing quotations.