System tray is a "legacy" tray where various applications (e.g. Nextcloud, Pidgin and Signal) have an icon with which you can interact with the application without actually opening the main application window. I said "legacy", because phasing it out was the plan in the Gnome 3 project, but it seems like we're not getting rid of the tray any time soon, if ever.
In any case the system tray is not enabled by default in recent Fedoras, and the proper way to enable it has changed several times in the recent past as Gnome has been upgraded. This outlines the most recent way to do it, and hopefully this one sticks for longer than a year or so.
First you need to install the AppIndicator Gnome shell extension. It is recommended to do that with dnf to ensure that all its dependencies are met and that it is always compatible with your current Gnome version:
sudo dnf install gnome-shell-extension-appindicator
This installs the extension but does not enable it. You may recall using Gnome Tweaks tool for enabling Gnome Shell extensions, but in Fedora 35 you would just get confused when the "Extensions" tab is completely missing. This is because management of extensions is now done with a standalone Gnome Extensions app, which can be installed like this:
sudo dnf install gnome-extensions-app
The final step is to launch the Gnome Extensions app:
Then enable the AppIndicator extension. Now you should start seeing your system tray applets on the right-hand side of your top pane.