A fatality is the final move done after you defeat your opponent and you hear the words "Finish Him!" on screen. You must press a series of specific buttons and your character will kill the other in a gruesome way.
This can be done in ANY game on any mode and it will probably be the first achievement you get.
Hit and go to 'Moves List' to see a full list of how to perform each fatality and where you must be standing.
Pick Sub-Zero and beat your opponent in both rounds. When you hear Finish Him, stand next to your opponent and press quickly. The screen will fade and Sub will perform the fatality.