Deleting a clip in the Clip Art pane removes it from your collections in the Clip Organizer. If it's an online clip that you made available offline (right-clicked it and then chose Make Available Offline), only the offline copy will be deleted. The online clip will still be available when you search with Include Bing content checked.
Tip: To tell whether a clip is in your offline collection, right-click it and choose Preview/Properties. Under Paths, look at Catalog. If the path is your computer (for example, starts with c:\users), the clip is in your collection. If the path is a URL (starts with http://), the clip is online.
Sometimes you'll see a message that a clip can't be deleted because it's in use. If you have the Clip Art pane open in another document, close the pane. Also try closing Clip Organizer if you have that open separately.