Apple - Move desktop item to system tray (dock)?
Firstly, @cyphorious has the answer you need, but I'm going to add more details to help expand your understanding of Mac terms.
Firstly, the .app file that is on your desktop is the actual application, not a link. Think of a .app as being roughly equivalent to a .exe (the main difference being that a Windows .exe file is a single file that will no doubt require a bunch of DLLs and other supporting files that it will expect to find either in your windows/system32 folder, or in the same location as the .exe itself. A .app looks like a single file, but is actually a collection of files and folders that is just treated differently so an app can be identified and moved/copied/deleted as a single file, rather than having to know which files and folders combined make it up - it's essentially a bit like a zip file that doesn't need unzipping to access the contents).
The tray of icons at the bottom is the dock. Nothing lives in the dock permanently, anything there should be a shortcut or link to the actual file that is expected to reside elsewhere. In this way, when you remove an app from the dock, you are just removing the shortcut, not deleting the actual app.
So the key to getting the shortcut working in the dock is to move the application to where you want it, then either start it and tell it to stay in the dock, or drag it to the dock from its new permanent location (usually the applications folder as per @cyphorious's answer, but could be on an external disk etc).
In OS X to "properly" install an application, just take the .app file a drag in into your applications folder using the Finder (due to localization the Application folder in the Screenshot is named Program..., but the icon is the same)
When you then launch the application it will appear in the dock (the "tray of icons"). Right click the icon and and choose "Options" -> "Keep in dock" and it will stay there for quick launching. You can also move it around in the dock if you want to have it in another spot.