How to close Skype down properly

You still have to right click on the little Skype icon in you tray area (lower right corner of the Windows screen, next to the digital clock) and select QUIT.

On a Mac check your Dock for the Skype icon:

If this is in your Dock, with the little bright dot beneath it, it is still running in the background.
Press Command (Apple) + TAB, letting go of tab but keeping Command (Apple) pressed down, a horizontal bar will appear in the middle of your screen, allowing you to cycle through your running apps when pressing TAB again. Move the rectangular cursor on the Skype Icon by pressing TAB repeatedly. Then, without letting go of the Command (Apple) key, press Q. This will close the Skype application.