Fatal Errors occur when either the parser of the evaluator cannot make sense of and expression you have typed in. They are presented in a popup message balloon under the equation editor and require you to edit your equation. Whenever possible, the portion of your equation which was not understood is selected for you, so you can easily spot the error and fix it. You don't need to interact with the error balloon (it will be hidden automatically when you fix the equation), but you can hide the message by pressing Esc or clicking the X in the upper-right corner of the messsage.
Warning Messages are errors encountered trying to evaluate specific point(s). By default they are not displayed, but you can toggle this with the Warnings Option"Print warning error messages" option in the General Settings tab of the Preferences dialog.
When enabled, warnings are shown as status bar messages.