Gutenberg is the new WordPress editor that will be released in WordPress 5.0, scheduled for release at the end of 2018. Have a look at how it works on WordPress.org.
When it was first released as a plugin for site owners to try out, it caused some major problems, including downtime. The plugin was a beta -- it was designed to find problems and was never designed to run on a live website. However, the fact that it broke sites has caused a lot of negative feeling, and many site owners want nothing to do with it.
However, Gutenberg has come along way from the initial development version release -- and the point of that initial release was to see what worked and what didn't, and to get bug reports so they could fix any issues before making a full release.
Unlike the initial release which came as an additional plugin, the full version of Gutenberg is going to be part of the WordPress 5.0 release. This means that it will be part of the default WordPress setup -- and the WordPress team will be working very hard to make sure it functions correctly.
In early August 2018, site owners started to see the option to "test the new editor".
No. Our recommendation is to not test the new editor on your live website.
But equally, you don't have to choose to commit to the classic editor like the notification seems to imply. We're expecting that the bugs will be mostly ironed out before that big release, and when it comes, we'll wait a week or so (as always) before installing it on sites on Performance Foundry hosting to see what errors arise. It’s going to be a complex upgrade, so we’ll be working very carefully to make sure all sites are upgraded effectively.
If Gutenberg not looking great at that point, we can consider the classic editor as an option, but making a decision now is unnecessary.
Just dismiss the notification and go on using your site as always. We'll look at Gutenberg again more closely when WordPress 5.0 comes out, and will be able to make a recommendation then about whether to move ahead with the full upgrade, or to choose the classic editor for a while longer.
(Performance Foundry is doing a lot of work behind the scenes to make sure our plugins and tools are compatible with Gutenberg and WordPress version 5. However, there's no need for your live site to be a test case!)