Android/iOS: Most two-factor authentication methods—which you should be using as much as possible—rely on codes from your phone to login to your account. Google’s simplifying it with a simple prompt on your phone that asks you to confirm you’re logging in.
This new feature will optionally replace your normal codes from the Authenticator app or SMS with a simple Yes/No prompt asking if you’re signing in somewhere else. To enable it, follow these steps:
Head to your account settings here.
Click Signing in to Google.
Click 2-Step Verification.
Enter your account’s password.
Scroll down to Google Prompt and click Add phone.
You’ll then be asked to answer the prompt on your phone as you would when logging in. This method is a lot simpler than typing in codes, but obviously it can have its downsides if you lose your phone. For that reason it’s probably a good idea to have SMS codes or another authentication app enabled just in case.
Google adds 2-step verification approval prompt on Android devices | Android Police