What are TeX and LaTeX?
First, Mike's answer is quite good. I will mostly expand on it and provide more details.
TeX
TeX is a language (a full programming language, actually) for typesetting documents. It originally output to a format called DVI which could then be converted to PostScript, PDF, etc.; more recent versions can output directly to PDF. You write a document with TeX instructions in it, and the TeX system will convert it into printable material.
TeX is used for a wide variety of documents, particularly in science and academia. Most people use it for things that other people would likely use Word for; however, the quality of its results are more on a par with InDesign or other major document layout packages, far superior what word processors generally yield. Designing specialized or ad-hoc document formats such as brochures, however, is probably easier with InDesign or QuarkXPress (although it is not impossible to do so in TeX/LaTeX).
TeX itself is quite low-level.
LaTeX
LaTeX is a macro package written in and for TeX that provides commands and defaults for writing larger documents at a higher level, taking care of things like sectioning, tables of contents, etc. In my experience, most TeX users do not write low-level TeX directly, but rather use LaTeX. LaTeX is not the only such package, though; ConTeXt is another macro package with a different design philosophy, but it sits at a similar level to LaTeX.
Usage
TeX and LaTeX are very widespread in some portions of academia, such as mathematics and computer science, due to its superb support for mathematical formulas. I have also heard that it is popular in some other disciplines as well, such as linguistics.
TeX is a typesetting language. You can use it to create high-quality PDF output. It's used a lot in the sciences because it handles equations very nicely. TeX is the base language; LaTeX is an extension of TeX that makes some common tasks easier. I’m sure someone else can explain some of the nuances better than I, as I’m still kind of green myself.
If you would rather use anything but Word, then TeX is for you.
"What is it used for?" - Typesetting every imaginable sort of document in commercial quality.
"Who uses it?" - Scientists, teachers, professors, musicians, illustrators, the Word-averse, technical writers, book publishers and more. I haven't used Word for any document (except when I've absolutely had to) since I started using TeX.
"What is it an alternative to?" - Word-processing programs, graphing utilities, graphics programs, etc
"How widespread is its use?" - To be honest, not that well-known beyond academia.
"Plain text or WYSIWYG?" - Generally, plain text mark-up. That's why we love TeX so much. That said, there are programs, such as LyX, that offer WYSIWYG with the power of TeX, so you could definitely check that out.