XFCE is a desktop environment similar to GNOME, Unity, and KDE (in idea, not execution/presentation). It is considered more lightweight than the others I listed, and also keeps to a more "traditional" desktop layout.
As to the applications that "use" it, I'm not sure how to answer that. There are of course an interconnected set of tools/applications that make up the base XFCE install, but no application uses it anymore that it would use any other desktop environment.
I was told to remove a bunch of XFCE related programs to fix an issue
with google-chrome, how can I know that I'm not using those programs?
Then this will remove XFCE, thereby effecting you if you are using XFCE. Otherwise, whatever desktop environment you're using is providing equivalent functions through other packages. I don't see why you would need to remove XFCE for Chrome, though, I use XFCE with Chrome (Chromium) just fine.