Basic Moves:

Moves that require only a button press:

HIGH PUNCH - Produces a punch to the face. Many can be strung together in succession by rapidly pressing the button.
LOW PUNCH - Your character will punch the opponent's midriff. Many can be strung together in succession by rapidly pressing the button. Not as powerful as a high punch but can hit the opponent while they are ducking.
HIGH KICK - Smacks your opponent in the face with a powerful kick
LOW KICK - Not as powerful as the high kick but a little faster to recover, kicks the opponent in the abs.
BLOCK - Blocks attacks 
RUN - Lets your character runs forwards for a limited amount of time. This is great for attacking opponents who are running away from you. Pay attention to the run meter (the smaller meter underneath your health bar) - this will indiciate how long you can run for.

Combination Moves:

Moves that require a button to be pressed simultaneously with a direction:

Jump Kick - Jump + either KICK button.
Jump Punch - Jump + either PUNCH button.
Pressing up plus kick or punch will result in your character punching or kicking in midair without moving in a direction. If jumping towards or away from the opponent, pressing punch or kick will cause your characrter to extend their limb a little further to hit the opponent and will cause more damage. A Jump Kick is more powerful than a Jump Punch, but Jump Pucnhes are better for winning aerial battles.
Uppercut - Down + HIGH PUNCH.
The most damaging normal move in the game - This will send your opponent soaring into the air.
Sweep - Back + LOW KICK
Sweeping knocks your opponent onto their back by tripping them over.
Roundhouse - Back + HIGH KICK
Your character spins around and gives your opponent a heavy kick to the side of the head. Like the Uppercut, this move is also very damaging.
Weak Kick - Down + LOW KICK
Not really useful unless your opponent is ducking and avoiding your other attacks...weak and slow.
Throw - Forward + LOW PUNCH
Must be standing right next to your opponent. Different characters throw in different ways, but all throws do good damage. Throws cannot be blocked normally.
Block Throw - BLOCK + diagonally down + away
Pressing down and away while blocking will prevent your opponent from executing a throw.
Death Punch - Standing punch when opponent is jumping towards you.
Timing is the key...this move is pretty difficult to pull off, but when you can do it, it is VERY damaging.
Power Hit - Forward + HIGH PUNCH
Some characters elbow, some headbutt, but the basic premise is the same - a strong hit to the opponent in close quarters. Most characters can perform a two-hit combo by pressing HIGH PUNCH twice. 

Special Notes:

To show you opponent Mercy, when it comes time to finish them, Hold RUN, press Down 4 or 5 times, and release RUN. Your opponent will be given back a small amount of health to continue playing.

Animalities can only be performed after winning the first round, losing the second, and winning the third after showing Mercy to your opponent.

Friendships and Babalities can only be performed after winning the first round, losing the second, and winning the third without using the BLOCK button. (Only the third round, not all three)

Stage Fatalities can only be performed on the Subway, the Bell Tower,  Scorpion's Lair and The Pit 3 stages.

Each character has their own unique combos, using a combination of punches, kicks, and sometimes blocks. Experiment to find the different combos for each character, I won't list them here because I'll probably miss some out!

Differences between the Arcade and Home versions:

Ultimate Mortal Kombat 3 had two different versions, one released for the arcade and Sega Saturn, and the other released for the Super Nintendo and Sega Genesis. The main differences between the two versions is extra playable characters and the addition of Brutalities.

In the arcade and Saturn version, there was only 4 extra playable characters over Mortal Kombat 3 (without the use of Kombat Kodes). These were Scorpion, Jade, Reptile and Kitana. Other characters were available through the use of Kombat Kodes.

In the Super Nintendo and Sega Genesis versions, Noob Saibot, Ermac, Classic Sub-zero, Mileena and Rain were selectable characters, in addition to the other 4 added in the arcade version. This version also introduced the new Finisher called the Brutality, which was pretty much a long combo that caused your opponent to explode at the end of it. The character Sheeva and Animalities were also removed.

For the sake of space, all characters that are also included in Mortal Kombat 3 will not be included in this guide (you can read the Mortal Kombat 3 guide for a complete list of their moves and fatalities) and only the new characters added in each version will be covered. However, for the SNES/Genesis version, I will include all of the character's Brutalities.

Arcade/Sega Saturn

Special Moves:

Moves specific to each character that must be done with a combination of buttons and directions.

Scorpion:

Spear: Back-Back-Low Punch
Teleport Punch: Down-Back-High Punch
Air Throw : Block

Fatality (jump distance): Down-Down-Up-High Kick
Fatality (close): Forward-Forward-Down-Up-Run
Animality (close): Forward-Up-Up-High Kick
Friendship (close): Back-Forward-Forward-Back-Low Kick
Babality: Down-Back-Back-Forward-High Punch
Stage: Forward-Up-Up-Low Punch

Kitana:

Fan lift: Back-Back-Back-High Punch
Fan Throw: Forward-Forward-High Punch+Low Punch together
Teleport Punch: Down-Back-High Punch

Fatality (close): Run-Run-Block-Block-Low Kick
Fatality (close): Back-Down-Forward-Forward-High Kicki
Animality (between close and sweep): Down-Down-Down-Down-Run
Friendship: Down-Back-Forward-Forward-Low Punch
Babality: Forward-Forward-Down-Forward-High Kick
Stage: Forward-Down-Down-Low Kick

Reptile:

Acid Spit: Forward-Forward-High Punch
Force Ball (Slow): Back-Back-High Punch+Low Punch together
Force Ball (Fast): Forward-Forward-High Punch+Low Punch together
Slide:  Back+Low Punch+Block+Low Kick simultaneously
Invisibility: Up-Down-High Kick
Super Elbow: Back-Forward-Low Kick

Fatality (jump distance): Back-Forward-Down-Block
Fatality (sweep): Forward-Forward-Up-Up-High Kick
Animality (close): Down-Down-Down-Up-High Kick
Friendship (close): Down-Forward-Forward-Back-High Kick
Babality: Forward-Forward-Back-Down-Low Kick
Stage: Block-Run-Block-Block

Jade:

Boomerang: Back-Forward-Low Punch
Boomerang (High): Back-Forward-High Punch
Boomerang (Low): Back-Forward-Low Kick
Returning Boomerang: Back-Back-Forward-Low Punch (will only return is opponent avoids boomerang)
Projectile Invincibility: Back-Forward-High Kick
Crunch Kick: Down-Forward-Low Kick

Fatality (close): Up-Up-Down-Forward-High Punch
Fatality (close): Run-Run-Run-Block-Run
Animality (close): Forward-Down-Forward-Forward-Low Kick
Friendship (anywhere): Back-Down-Back-Back-High Kick
Babality: Down-Down-Forward-Down-High Kick
Stage: Back-Forward-Down-Run

Secrets:

Kombat Kodes (see the Mortal Kombat 3 FAQ for explanations of the Kombat Kodes):

999-999 - Show what revision you have.
010-010 - Throwing Encouraged (displays funny message when you throw)
100-100 - No Throws
020-020 - No Blocking
987-123 - No Meters (all meters at the top are hidden)
300-300 - Silent Kombat (no music)
788-322 - Quick Uppercut Recovery (no lag after you uppercut)
033-000 - Half Energy for Player 1 (start with 1/2 energy every round)
000-033 - Half Energy for Player 2 (start with 1/2 energy every round)
707-000 - Quarter Energy for Player 1 (start with 1/4 energy every round)
000-707 - Quarter Energy for Player 2 (start with 1/4 energy every round)
044-440 - Sans Power (starts game on round 3 with almost no energy)
688-422 - Dark Fighting (screen lights up when you hit)
444-444 - Randper Kombat (random morphing) (or 460-460)
985-125 - Psycho Kombat (dark,randper,no meters,no block,quick uppercut)
466-466 - Unlimited Run (run meters don't go down after combo/run)
642-468 - Galaga (game over for both players after you play)
091-190 - Bell Tower
077-022 - Bridge
933-933 - Ermac's Portal
666-333 - Graveyard
330-033 - Jade's Desert
004-700 - Kahn's Kave
880-220 - Kahn's Tower
600-040 - Kombat Temple
050-050 - Noob's Dorfen (Bell Tower)
820-028 - Pit III
002-003 - River Kombat
343-343 - Roof
666-444 - Scorpion's Lair
123-901 - Soul Chamber
079-035 - Street
880-088 - Subway
717-313 - Rain can be found at the graveyard (Text message - no effect on gameplay)    
448-844 - Don't jump at me - MXV (Text message - no effect on gameplay)
122-221 - Skunky!! - E.F. (Text message - no effect on gameplay)
004-400 - Watcha gun do ? - E.B. (Text message - no effect on gameplay)
550-550 - Go see Mortal Kombat the live tour! (Text message - no effect on gameplay)
282-282 - No Fear (Text message - no effect on gameplay)
123-926 - No Knowledge that is not power (Text message - no effect on gameplay)
987-666 - Hold Flippers during casino run (Text message - no effect on gameplay)
969-141 - Winner Fights Motaro
769-342 - Winner Fights Noob Saibot
033-564 - Winner Fights Shao Kahn
205-205 - Winner Fights Human Smoke
227-227 - Explosive Kombat (2 on 2 mode only)
022-220 - Explosive Kombat and Throwing Disabled (2 on 2 mode only)

Supreme Kombat Kodes:

These are entered after you beat Shao Kahn. How many boxes you have to choose from depends on what skill level you finished the game on.

Novice:  1-4
Warrior: 5-6
Master:  7-8
Master:  9-12 (The higher Master level)

1  MK Dragon      : See the character's ending
2  MK Logo        : Galaga
3  Yin-Yang       : Fight Ermac
4  Number three   : Fight Noob Saibot
5  Question Mark  : Random select (selects another code)
6  Lightning Bolt : Fatality Demonstration #1
7  Goro           : Fatality Demonstration #2
8  Raiden         : Fatality Demonstration #3
9  Shao Kahn      : Fight Noob and Ermac
10 Skull          : Fight Classic Sub-Zero and Noob
11 Question Mark  : Fight Noob,Ermac,Human Smoke,Classic Sub-Zero,Mileena
12 Question Mark  : Supreme Demonstration

Super Nintendo/Sega Genesis

Mileena:

Sai Throw: hold High Punch 2 seconds, release
Teleport Kick: Forward, Forward, Low Kick
Roll Attack: Back, Back, Down, High Kick

Fatality:  Down, Forward, Down, Forward, Low Punch (close)
Fatality:  Back, Back, Back, Forward, Low Kick (far)
Friendship: Down, Down, Back, Forward, High Punch
Babality: Down, Down, Forward, Forward, High Punch
Stage: Down, Down, Down, Low Punch
Brutality: High Punch, Low Punch, Low Punch, High Punch, Block, High Kick, Low Kick, High Kick, Block, High Punch, Low Punch

Noob Saibot:

Katatonic Blast: Down, Forward, Low Punch 
Mirror Image Throw: Forward, Forward, High Punch
Teleport Slam: Down, Up

Babality: Forward, Forward, Forward, Low Punch 
Stage: Down, Forward, Block
Brutality: High Punch, Low Kick, Low Punch, Block, Low Kick, High Kick, High Punch, Low Punch, Block, Low Kick, High Kick

Noob Saibot had no Fatalities or Friendship of his own in this version.

Ermac:

Telekenetic Slam: Back, Down, Back, High Kick 
Teleport punch: Down, Back, High Punch 
Fireball: Down, Back, Low Punch

Fatality: Run,Block, Run, Run, High Kick
Fatality: Down,Up, Down, Down, Down, Block (1/3-2/3 screen)
Stage: Run, Run, Run, Run, Low Kick
Brutality: High Punch, High Punch, Low Punch, Block, High Kick, Low Kick, Block, High Punch, Low Punch, Low Kick, High Kick

Ermac had no Babality or Friendship in this version

Classic Sub-Zero:

Ice Blast: Down, Forward, Low Punch 
Ice Puddle: Down, Back, Low Kick 
Slide: Back+Low Punch+Block+Low Kick simultaneously

Fatality: Down, Down, Down, Forward, High Punch (close)
Stage: Forward, Down, Forward, Forward, High Punch
Brutality: High Punch, Low Punch, High Punch, Block, Low Kick, Low Kick, High Kick, High Kick, Low Punch, High Punch, Low Punch

Classic Sub-Zero has no second Fatality, Babality or Friendship in this version.

Rain:

Lightning Strike: Back, Back, High Punch 
Force Ball: Down, Forward, High Punch (can control opponents after connecting) 
Super Roundhouse: Perform a normal roundhouse, but Rain's sends the opponent flying off one side of the screen and landing on the other side.

Babality: Forward, Back, Back, High Punch 
Stage: Forward, Down, Forward, Low Kick
Brutality: High Punch, High Punch, Block, Low Kick, High Kick, Block, Low Kick, High Kick, Block, Hight Punch, Low Punch

Rain has no Fatalities or Friendships in this version.