How do I set up a dropdown navigation?

To add dropdown navigation to your store's main menu from your Shopify admin open: Online Store > Navigation 
Here you are presented with all of the menus that exist within your store, identify the menu that you would like to work with. 

By default in a new Shopify store, the primary navigation will be named: Main Menu click Edit next to your chosen menu.
A standard menu with dropdowns will look like this:

Note how Shop contains a submenu - this will reflect in your store's live view. When Shop is hovered its submenu will appear.

To create this you will click on Add menu item to create a new link. When added simply drag that link to beneath it's parent item and drag slightly to the right.
This will create a parent > child relationship between those links allowing for the creation of a submenu.

The following video illustrates the process for creating submenus within a Shopify menu: