Note: All gifs were made from the Genesis version of UMK3. As per usual, my disclaimer of the gifs not accurately portraying the game's animation, speed or framerate apply here. The gifs have also had their color palette reduced to help out those with modems.

Also, please note that the Genesis and SNES versions of UMK3 did not feature animalities, and some characters didn't even have a single fatality.
None
(in the Genesis/SNES versions)
Fatality - Tongue Rip
Fatality - Acid
Friendship - Reptile in the Box
Fatality - Body Whipper
Fatality - Impaler
Friendship - Pogo Stick
None
(in the Genesis/SNES versions)
Originally nothing but a rumor in MK3, Boon and Tobias perpetuated it by programming in a Kombat Kode that would put the following text on the screen before your fight: "Rain Can Be Found in the Graveyard". In MKT, they actually created and added him to the MK story.
Reptile's Tongue Rip fatality comes back again in UMK3...and with quite a twist.
Jade is one of the few characters that attacks with a weapon in her combo. Others include Scorpion and his axe, Kabal and his hook swords, and Mileena and her sias.
Noob Saibot...for those who haven't figured it out, is actually just Boon Tobias backwards. He originally appeared in MK2, and could be fought if you won 50 matches in a row.
Fatality - Nail Spitter
Fatality - Bone Storm
Friendship - Mirror, mirror...
Fatality - Spine Tear
Fatality - Kiss of Death
Fatality - Fan Chop
Friendship - Bubbles
Fatality - Telekinesis Slam
Fatality - Super Uppercut
The Nail Spitter Fatality...what in the FUCK were they thinking with that one!?
What you miss just by seeing the gif is the "ripping" noise that comes soon after the screen blacks out. It's sort of trying to convey that SZ is ripping off someone's head like he did in MK1.
Becky Gable plays Jade, Mileena and Kitana(gasp!).
Ermac's Super Uppercut fatality was the spark that let loose the various rumors that he was Johnny Cage in disguise. Upon MKT's roster announcement, those rumors were finally proven false.
Scorpion's Lair
Fatality - Toasty
Fatality - Hellfire
Friendship - Skull in the Box
Brutality
Sort of a half-assed, yet kinda hip stage-fatality.
In the arcade version, Scorpion had a different fatality that was never ported to *any* home version of the game. Basically, it's a lot like the "Hellfire" fatality of the Genesis version, but rather than having the opponent burst into flames, Scorpion summons about 20 clones of himself that all rise out of the ground, then run towards the opponent as the screen blacks out. It was very cool.
One of the coolest -alities ever. You basically hit a series of buttons, then the character takes it from there. They beat the living fuck out of the opponent until they blow up.