Download the full MacTeX package from here. This is a very large file, 2.2 GB, and will take several hours to download. It contains both the front and back ends, as well as many optional extras. Once you have downloaded the file, open it and you will be guided through the installation. This will place TeXShop in the folder Applications/TeX/.
Download MiKTeX from here. There are several versions of MiKTeX for the various version of windows. Follow the installation wizard instructions; the default values should work for everyone. Once installed, navigate to the MikTeX folder under programs and run TexWorks to start writing beautiful documents!
For Linux (Ubuntu/Debian)
LaTeX can be run solely on the command line. To do this, a package called texlive is required. This is included by default in almost all distributions. However, there are lots of useful extra packages, so a package called texlive-full is recommended. This is a fairly large package (> 1 GB). Install it by running ‘sudo apt-get install texlive-full’.
If you just want to use the terminal, a .tex file can be typeset, i.e. compiled to pdf, with the simple command 'pdflatex filename.tex'. You can create the .tex file in your favorite plain text editor (vim, emacs, gedit, and sublime text all have basic syntax highlighting for LaTeX).
Several IDEs are available for LaTeX. These generally make it faster and easier to use LaTeX. We recommend Kile. To install Kile, just run 'sudo apt-get install kile' or use your favorite method of installing packages.