Xubuntu Core is a slimmed down version of Xubuntu that doesn’t come with many of the applications, plugins, and libraries that the default Xubuntu desktop ISOs have. It essentially only ships Xfce apps (window manager, panel, file manager) and the basic look and feel of Xubuntu, so there will be no browser, email client, text editor, file search, office suite, image or media player, package or update managers, community wallpapers, etc, etc.
The obvious benefit of a Xubuntu Core ISO will be faster downloading – especially interesting for those with limited connectivity – and you can then add only what you want.
Since Xubuntu 14.10 Utopic, it’s available as a task and metapackage. The goal of this is not to create a lightweight version of Xubuntu, but more targeted towards advanced users to give them a base to work off of. See also Appendix A of the Xubuntu docs.
Does this have anything to do with Snappy, Ubuntu Core, or Convergence?
Nope, not a dang thing.
Official Installation Method
The current official method is to download the mini.iso, install, and when prompted, install the 'Xubuntu minimal installation' task. If you’d prefer to wait until after the installer finishes to install the Xubuntu core task, you can simply enter
sudo apt-get install xubuntu-core^. Don’t forget to add the caret so that the xubuntu-core task instead of the metapackage is installed.
Of course, if you won’t have internet during installation, or don’t want to mess with the mini.iso, tasks, and the Debian installer, you can install from the Xubuntu Core ISO on the downloads page.