sudoers - simple explanation requested

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http://askubuntu.com – Everytime I want to be able to run something that requires me to be a sudoer too many times, I need to google for the formatting of /etc/sudoers to remind me again what exactly is the proper way to write it. Now I see different writing styles in my sudoers file, which is the consequence of different google results over the months. I've also noticed that the second example (below) seems to work in XFCE, but not in Cinnamon (Gnome 3). This could be totally unrelated, but nontheless I'd like to know once and for all, what is the correct grammar of the sudoer line, and what is the difference betw (HowTos)