I do not currently teach in a classroom of my own as I am a substitute, but I can see many ways in which a teacher might utilize Gmail. For one, Google Hangouts could be a quiet way for students to chat with one another in the classroom if they have finished a test early and do not want to disturb others. Hangouts could also be used for a homework group, to share what assignments are due if a student is absent or to keep track of one another's progress.
I personally use Gmail filters to keep my family contacts in one place so I can email all of them at once with important news. I also plan to use the "family" label to keep track of conversation threads between my siblings, parents and myself.