Once upon a time, there was an operating system where everything ran smooth, there weren't any errors, issues or omissions, and everyone used the same happy-go-lucky device. Well, that's how a fantasy novel about mobile operating systems should start anyway, with Android in the role of the villain.
Among the myriad of issues you can encounter among the myriad of Android devices out there, is the dreaded "Authentication is Required. You need to sign into your Google Account." error message that keeps popping up in your notification shade when you, say, try to download and install an app from the Play Store.
We won't be going into details as to why this is happening even when you have signed into your Google Account, or created a new one from scratch when logging into the phone, but will just tell you how this annoyance might be fixed quickly:
Official Google fix
As per Google's own advice
on the matter, you need to go into your phones Settings > Apps or Application Manager list, scroll to find the Google Play Store app (they are arranged alphabetically), and clear the Play Store's data with the respective button;
Logging in with your Google Account from the Play Store itself
1. When the previous one fails (and with us it did), try going to Settings > Accounts > Google, and delete the problem account. Be sure that you know the account password so you can add it again;
2. Now, don't re-add your Google Account from the actual Account menu in Settings, but open the rogue Google Play Store app on your phone or tablet, and when prompted for your user name and password, enter those inside the Play Store itself.
That's all, your app downloading and installation should be working now, and a new account will be added in your list in the Settings menu, though you may still need to allow it individually the first time you enter apps and sites where your Google credentials are needed.