Google is apparently testing a new feature that would allow you to securely sign into your Google account on the web without needing to use your password. Once set up a compatible phone, such as a Nexus 6P, users can apparently authenticate directly from that phone.
The test seems to be extremely limited. Rohit Paul posted about his experience with the test on Reddit:
You authorize your phone to allow you to log into your account. You go into a computer and type in your email. Then you get a message on your phone to allow the login. If you hit yes, the computer logs into your Google account without a password.
Google first unveiled this feature as part of the Marshmallow API during Google I/O 2015. It's very similar to the existing Smart Lock system that the company has in place for Chromebooks, which allows a user's phone to unlock a Chromebook over Bluetooth.