The Desktop is a place to save files to and view mounted volumes/drives/USB sticks.
As Coconut battery is a ZIP file, that may have been already extracted by the browser of your choice, it will only show up as a file on the Desktop (or the Downloads folder).
When you start the actual Coconut application, from the Desktop or the Downloads folder, an icon will appear on the Dock, to show, that this program is running.
The Dock is not an actual place to put files to, just a placeholder for aliases/shortcuts to applications or folders and the Trash.
Putting a running application into the Trash is not recommended. And running an application, that sits in the Trash, is not possible, as you experience.
Just put the actual Coconut app into the Applications folder, and it won't show up on the Desktop, just in the Dock.
If you mean the window, where all the stats are, you can close the window by clicking on the red circle on the top left, but that behaviour also quits this program.
Was that of any help?
Maybe this can help you:
http://www.apple.com/findouthow/mac/