Google is a very security conscious company and always working on ways to keep users safe. They are currently testing a new way to sign into your Google account without needing to enter a password. One user on Reddit was invited to test the new method and posted detailed screenshots of the process.
If launched, users will be able to activate the feature by going to the Sign-in & security section of their Google account settings. Users will need to first associate a primary smartphone with their Google account and likely make sure their phone is locked via PIN or password.
When a user wants to access their Google account on the web, a notification will be sent to their phone asking if they’re trying to sign-in. Clicking yes will refresh the page on the web and log-in. Users will still be able to sign-in with their password at any time or in the scenario that their phone runs out of battery. Occasionally, there will be a challenge consisting of entering a number listed on the web into your phone.
This password-less entry is convenient for people who easily forget passwords and as such opt for easy, insecure ones. It is also useful for when you need to log-in from a public computer. This way your actual password never gets transmitted or needs to be entered. The notification that pops up on your phone is likely a feature of Google Play services. This user likely got access to the feature as a part of random testing that Google frequently does, as such it is unknown if Google plans to roll this out to all accounts.