Following a student means you are automatically CC'd on any comments related to the student, sort of like following someone on Twitter. Followed students are also included on Mastery and Missing Work reports in your Teacher Dashboard "Other Reports" tab.
To select students to follow, simply access your Follow List from the Account & Support Menu:
From there, simply type in the names of the students you'd like to follow. If you'd like to "Follow All" students, click on the options menu in the top right and select "Follow All."