1) Download Processing 2
You will first need to download the latest version of Processing. Choose the appropriate download for your operating system.
2) Install Processing 2
Once your download has completed unzip Processing and copy the folder to the place where you usually put software on your computer. In Windows this will be the Program Files directory.
3) Setup Processing
Double-click on the Processing executable file, Processing.exe, that's located inside your Processing directory. When Processing runs for the first time it creates a directory on your computer called the sketchbook - this is where all your processing sketches will be stored. You can accept this default location, or you can move the sketchbook to the place where you usually store your personal files (for example a Dropbox).
To move the sketchbook first create a folder called 'sketchbook' in the location where you'd like the folder to be, then change the link to your new sketchbook in Processing under File>Preferences. You then need to close and re-open Processing for this change to take effect.
Your life will be much easier if you associate files with the .pde extension with Processing. This means that when you double-click on any Processing files it will automatically open in Processing. You can do this in Windows by right-clicking on any Processing file (for example one of the .pde files in your sketchbook) and choosing Open with > Select default Program. If you can't see file extensions you should turn this on by following the instructions here - this also includes instructions for setting the default application for a given file type.
4) Restart Processing to complete the installation