Google use a rather clever single sign-on system for all their products like Gmail and Google Calendar (the free ones and the paid-for G Suite ones). If you only have one account (e-mail address) it’s simple enough, but with multiple accounts it can be a bit confusing until you get the hang of it. Here’s a quick guide.
Go to the Google search page and check the top-right of the page:
then the PC user is already logged in (with this web browser) . Click on the coloured circle and then on Add account:
In some circumstances the above window may show multiple accounts (if you want to use one of these, click on it) but you can still use Add account:
You will then see the log in screen to add another account:
In some circumstances you may see an intermediate screen:
In this case, just click Add account again.
It is also possible to add an account from within Gmail or other Google application, again by clicking the circle:
If you use multiple web browsers (Edge, Internet Explorer, Chrome, Firefox, Safari, etc) each browser maintains its own list of Google accounts that you are logged into. On each browser, logging out of one account logs out of all of them.