LaTeX is a macro package for DonaldKnuth's TeX typesetting language. See 
http://www.latex-project.org/ for more information.
Rather than specifying the exact layout of a document, for example fiddling with fonts and margins in a WYSIWYG WordProcessor such as MicrosoftWord, you specify the structure of the document. (A bit like what HTML was supposed to be like before the internet became commercialised. :-) ) It offers much greater control over layout and appearance than other WordProcessors.
Basically, people a lot smarter than you have come up with good formatting guidelines for different document types, so you don't have to worry about that sort of stuff. You just write something like
\section{Section Heading}
In a \emph{technical report}, you should always have some text between
a heading and a sub heading, so here is a sentence.
\subsection{First Subsection}
Blah blah blah...
LaTeX automatically keeps track of section and figure numbers, and comes with packages to help keep track of bibliographies/references. Source files get processed and can be converted into popular formats such as PostScript and PDF (using pdflatex(1)).
You can automatically mark up plaintext files and SourceCode as LaTeX. This used to be done by the lgrind program, but that is non-Free software. There is a Free LaTeX package called listings that does the same thing and should be included with your LaTeX distribution (on Linux, at least).
LyX is a WYSIWYG LaTeX editor, or something approximating it.
To learn how to compile documents with LaTeX, see LatexMakefiles.
To follow through a quick worked example on using LaTeX, see LatexExample.
Because LaTeX is a macro language, it's hard to get an accurate word count. See LatexWordcount for some tips.
There are several packages that you can run to find common typos and simple LaTeX errors in your files; try installing the chktex package (dev-tex/chktex on Gentoo). (For example, it will tell you where you should add non-breaking spaces, or suggest where you've used the wrong type of dash.)
Also see PdfLatexNotes, and the ManPage, latex(1)
(The correct spelling is LaΤεχ, but it is neither a WikiWord, or easy to type.)
