Simple text copy-paste functionality is provided for free by widgets such as Gtk::Entry and Gtk::TextView, but you might need special code to deal with your own data formats. For instance, a drawing program would need special code to allow copy and paste within a view, or between documents.
You can usually pretend that Gtk::Clipboard is a singleton. You can get the default clipboard instance with Gtk::Clipboard::get(). This is probably the only clipboard you will ever need.
Your application doesn't need to wait for clipboard operations, particularly between the time when the user chooses Copy and then later chooses Paste. Most Gtk::Clipboard methods take sigc::slots which specify callback methods. When Gtk::Clipboard is ready, it will call these methods, either providing the requested data, or asking for data.