A shortcode is a WordPress-specific code that lets you do many things with very little effort. Shortcodes can embed files or create objects that would normally require lots of complicated, ugly code in just one line. Shortcode = shortcut.
You can also add shortcodes with the regular text editor widget, but the shortcode widget is much more powerful because you can see how the shortcode looks like without going to preview mode. It is rendered automatically.
The following is a GIF video to show you how to use the shortcode widget :
There are many uses for shortcodes: Adding Contact Form 7, audio widget or any website API.
A shortcode, basically, is a special tag that you enter into a page on WordPress, that gets replaced with different content when you view the live website.