You haven't said what version and operating system.
If you have the computer set to access your emails through a email programme - Thunderbird (mac or pc), Apple Mail (mac), Windows Mail (pc) etc., then you need to remove the account settings from the programme.
If you access your email via webmail - then you need to sign out of it in the browser.
It would alos be a good idea to change your passwords - so that if the family member came across any of your passwords, they'd be no use to them.