I will not be able to give any better explanations than www.xm1math.net/doculatex/install_miktex.html (in French, TODO find in English). They are complete, trustworthy, and well written.
I recommend TeXMaker to write \(\LaTeX{}\) on your Windows computer : http://www.xm1math.net/texmaker/download_fr.html#windows (in French, TODO find in English). Currently, the direct link is : http://www.xm1math.net/texmaker/texmakerwin32_install.exe (55 Mo).
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On Linux ?
The usual text editors (emacs, Sublime Text 3, vi(m), Geany, Kate, or even the minimalist GNU Nano) are all doing a pretty good job (they might require to install one additional plugin, e.g. LaTeX-plus for Sublime Text 3).
For a editor specific to \(\LaTeX{}\), TeXMaker is also available on Linux, but I recommend the awesome Kile editor.
That list (on Wikipédia) can help you choose a good editor to write awesome \(\LaTeX{}\) documents.
And I also recommend the MikTeX \(\LaTeX{}\) "distribution" : MikTex.org/2.9/setup.
Currently, the direct installer link is: http://mirrors.ctan.org/systems/win32/miktex/setup/basic-miktex-2.9.5105-x64.exe (160 Mo).
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Sur Linux (2) ?
On both Debian and Ubuntu, the recommended distribution is TexLive, which is easily installable thanks to apt-get (or pacman, or .. whatever your paquet manager is).
If any issue occurs during the installation... do it again by following step-by-step the official tutorial.
To learn, or refresh your memory about \(\LaTeX{}\), the Wikibook en.WikiBooks.org/wiki/LaTeX will be more than useful.
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Examples of maths written with \(\LaTeX{}\)?
Some of my pages embed a demo: