"Third-party authentication" means accessing a database using, say, a Google account. I've been trying it out with several of our clients since the start of the summer. Since this was a new feature introduced in May with FileMaker 16, it's taken time to evaluate, but I've finally come to some conclusions. Executive summary: third-party authentication is potentially safer than a native FileMaker account and password. But whether it's actually safer or not, depends on the particular Google account.
I've written a new article going into this in greater depth in our support library. You can read it here: "The pros and cons of third-party authentication."
This has led to a change in my user sign-up process, effective immediately. I used to ask users if they wanted to sign in using Google or a local FileMaker account. I'm not asking any more. All new users now get local FileMaker accounts to start with. If you want to use your Google account to sign in, contact me.