I recently struggled with the dilemma of selecting the right icon for my navigation menu, after an update to one of my apps.  After some googling, I was surprised to find out that even Google Chrome has flip flopped around with this, from the hamburger, to wrench, and then to dots.  After briefly exploring, here are my 2 cents on the matter.

The Hamburger aka bars

The most appropriate use for the hamburger is to denote a full screen menu or a side menu.  The hamburger is often seen in mobile applications and has become common enough that it needs no further explanation.

The Vertical Dots aka options, vertical elipsis

This icon is used for "more options" that cannot fit on your navbar, and usually takes the form of a dropdown menu.  This one seems less common outside of Google apps, and one could argue that the icon alone is not very intuitive.  I'd recommend only using this icon at the far right of a navbar.  All other dropdown menus outside of a navbar should use the down chevron or down caret.

The Wrench or Screwdriver aka tools

This is more of a legacy icon, evolved from old Windows or Mac applications, and used to denote a menu of tools.  This icon can be used for a menu or a dropdown, so long as it displays a selection of tools (e.g. print, save, save as, etc).

The Cog aka gear

Yet another legacy icon that is still very commonly used today.  It goes without saying, that this icon is used for settings.  This icon commonly appears in a navbar, and could be used to open as a menu or a dropdown, but it should be reserved for a selection of settings only (e.g. volume, brightness, zoom, etc).

Final Thoughts

I'm far from a UI/UX expert, and am instead just a guy who probably thought way too hard on this subject matter.  What are your thoughts?  Feel free to comment below.  

Ultimately, you should select the icon that you think is best for what you're trying to achieve.  If you get stuck, try asking and testing with a few friends.  And, one final tip for if you're just starting out - take some time to learn how to use font awesome, which will make your life a lot easier.