Following sites is one of the new features of Sharepoint 2013. It allows users to track sites which are most interesting or relevant for them. In order to follow the site user should click Follow link in top right corner:
It will show message that current site is now followed:
In order to unfollow the site you need to click on Sites link near Office 365 in top left corner:
Here you will see list of all followed site. Near each site there will be context menu which will allow you to unfollow appropriate site:
After you will click “Stop following” site will disappear from the list of followed sites.