The front end of your website is what your site visitors see.
The back end is what the site administrators and editors see, AKA the WordPress Dashboard.
Frequently these terms are used to describe skill sets, such as a front-end developer vs. a back-end developer. The terms are rather fuzzy, and certainly overlap a lot, and many people have skills from both of the following skill sets.
A back-end developer usually deals with customizing plugins, PHP code, MySQL, custom post types, and site functionality.