How do I sign in to multiple Microsoft accounts in the same web browser?

What I do at work is use Incognito mode in Chrome for my personal accounts.

Just hit Ctrl+Shift+N from an open Chrome window, or you can find the option in the browser menu. Firefox and IE have a similar feature (Private window and InPrivate, respectively).

The OP didn't ask specifically for Chrome, so if you're OK with Firefox, you can use the Test Pilot feature called "Container Tabs." Works great to sign in to multiple accounts from the same (or another) Firefox window.

See https://testpilot.firefox.com/about/ for details on Test Pilot and https://testpilot.firefox.com/experiments/containers for the Containers experiment.