Fatalities are the final moves done at the end of each battle by the winning fighter. After the announcer shouts "Finish Him!" you must perform a specific combination of buttons to do the fatality. Distance is a key factor. Press and select Move List to see how to perform each Fatality or read this thread for a complete move list. If the Fatality requires you to press up on the d-pad, hold the Block button while inputting your combination so you don't jump.
You can keep track of this roughly in the Nekropolis by checking how many "-alities" a specific character has (beware, this also counts babalities), by keeping a checklist, or by running through all the characters in order. The Sans Power code again speeds things up.
This can not be done in Fatality Trainer.