Update December 19, 2016: Updated to reflect changes to Google's policies and user interface.
Even if your student doesn't use a Chromebook, they may have a Google Apps for Education (GAFE) account, which is essentially a Google account provided by the school. GAFE accounts can be accessed from any browser, and the privacy settings affect your student regardless of which browser they use to log in to GAFE. In order to maximize your child's privacy, we advise changing the settings below.
If your child's school issued them a Chromebook, be sure to also check out our Guide to Chromebook Privacy Settings for Students.
Using any web browser, open the URL https://myaccount.google.com/activitycontrols and make sure you are signed in to your Google account. Then, make sure all the slider/toggles are to the left ("paused"). If a confirmation box pops up asking you to confirm that you want to pause an activity, confirm it by clicking "Pause". Additionally, under "Web & App Activity", uncheck "Include Chrome browsing history and activity from websites and apps that use Google services." (Also, note that depending on your school's settings, not all of the slider/toggles may be available for your account.) 
Congratulations! You've successfully enabled the most privacy-protective settings in your child's GAFE account!
And don't forget, if your child's school issued them a Chromebook, be sure to also check out our Guide to Chromebook Privacy Settings for Students.
