You start Mortal Kombat X with one Fatality and one Brutality for each default character. You must discover the remaining Fatalities and Brutalities by performing them at the end of a match or unlocking them in the Krypt. If you perform a Fatality before you unlock it in the Krypt, it will automatically unlock. However, Brutalities must be unlocked before you can perform them in a match. You can also perform Fatalities in the Fatality Practice mode before they've been unlocked. Just perform the new Fatality instead of the one listed on-screen.
Fatalities must be performed from a specific distance away from your opponent:
Note: In some cases it's easier to hold Block while performing the directional inputs for a Fatality. For example, if the Fatality if Forward, Back, Forward, 3 it may be easier to hold Block, press Forward, Back, Forward, release Block, then press 3.
Brutalities are essentially special moves that perform a Fatality when used as the final hit that kills the opponent and ends the match (before the announce yells, "FINISH HIM".) However, to perform most Brutalities you have to meet certain requirements. For example, you may have to hit the opponent a certain number of times with an attack during the match, win the first round of the match with a throw, or hold Forward during the final attack animation. If all requirements are not met, the Brutality will not happen.