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, 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!

Special Moves:

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

Sonya:

Force Wave: Down, Forward, Low Punch
Leg Grab: Down + Low Punch + Block
Teleport Punch: Forward, Back, High Punch
Bicycle Kick: Back, Back, Down, High Kick

Fatality ( Further then half screen ): Hold Block + Run, Up, Up, Back, Down, release Block+Run
Fatality: Back, Forward, Down, Down, Run
Animality ( Close ): Hold Low Punch, Back, Forward, Down, Forward, release Low Punch
Friendship: Back, Forward, Back, Down, Run
Babality: Down, Down, Forward, Low Kick
Stage: Forward, Forward, Down, High Punch

Nightwolf:

Arrow: Down, Back, Low Punch
Hatchet Uppercut: Down, Forward, High Punch
Shadow Charge: Forward, Forward, Low Kick
Reflect Shield: Back, Back, Back, High Kick

Fatality ( close ): Up, Up, Back, Forward, Block
Fatality ( Full Screen ): Back, Back, Down, High Punch
Animality ( Close ): Forward, Forward, Down, Down
Friendship ( Sweep Range ): Run, Run, Run, Down
Babality: Forward, Back, Forward, Back, Low Punch
Stage: Run, Run, Block

Kung Lao:

Hat Throw: Back, Forward, Low Punch
Teleport: Down, Up
Torpedo Kick ( In Air ): Up, Down, High Kick
Spin Attack: Forward, Down, Forward, Run ( press Run several times )

Fatality: Run, Block, Run, Block, Down
Fatality ( Between Sweep Range and Close ): Forward, Forward, Back, Down, High Punch
Animality ( Close ): Run, Run, Run, Run, Block
Friendship ( Sweep Range ): Run, Low Punch, Run, Low Kick
Babality: Down, Forward, Forward, High Punch
Stage: Down, Down, Forward, Forward, Low Kick

Kano:

Knife Throw: Down, Back, High Punch
Knife Uppercut: Down, Forward, High Punch
Cannon Ball: Hold Low Kick for 4 seconds, release
Neck Choke: Down, Forward, Low Punch
Air Throw ( In Air ): Block

Fatality ( Close ): Hold Low Punch, Forward, Down, Down, Forward, release Low Punch
Fatality ( Sweep Range ): Low Punch, Block, Block, High Kick
Animality ( Close ): Hold High Punch, Block, Block, Block, release High Punch
Friendship: Low Kick, Run, Run, High Kick
Babality: Forward, Forward, Down, Down, Low Kick
Stage: Up, Up, Back, Low Kick

Cyrax:

Close Grenade: Hold Low Kick, Back, Back, High Kick (releasing Low Kick not necessary)
Far Grenade: Hold Low Kick, Forward, Forward, High Kick (releasing Low Kick not necessary)
Net: Back, Back, Low Kick
Teleport ( can be done in the air ): Forward, Down, Block
Air throw ( you on ground opponent in air ): Down, Forward, Block Then Low Punch to throw.

Fatality : Down, Down, Up, Down, High Punch
Fatality ( Close ): Down, Down, Forward, Up, Run
Animality ( Close ): Up, Up, Down, Down
Friendship: Run, Run, Run, Up
Babality: Forward, Forward, Back, High Punch
Stage: Run, Block, Run

Liu Kang:

High Fireball: Forward, Forward, High Punch
Low Fireball: Forward, Forward, Low Punch
Flying Kick: Forward, Forward, High Kick
Bicycle Kick: Hold Low Kick for 5 seconds, release

Fatality: Forward, Forward, Down, Down, Low Kick
Fatality: Up, Down, Up, Up, Block + Run
Animality ( Sweep Range ): Down, Down, Up
Friendship: Run, Run, Run, Down + Run
Babality: Down, Down, Down, High Kick
Stage: Run, Block, Block, Low Kick

Kabal:

Dash: Back, Forward, Low Kick
Eye Spark ( can be done in the air ): Back, Back, High Punch
Ground Saw: Back, Back, Back, Run

Fatality ( Sweep Range ): Down, Down, Back, Forward, Block
Fatality ( Close ): Run, Block, Block, Block, High Kick
Animality ( Close ): Hold High Punch, Forward, Forward, Down, Forward, release High Punch
Friendship ( a bit further then sweep ): Run, Low Kick, Run, Run, Up
Babality: Run, Run, Low Kick
Stage: Block, Block, High Kick

Stryker:

High Grenade: Down, Back, High Punch
Low Grenade: Down, Back, Low Punch
Baton Sweep: Forward, Back, Low Punch
Baton Toss: Forward, Forward, High Kick

Fatality ( Close ): Down, Forward, Down, Forward, Block
Fatality ( a bit closer then full screen ): Forward, Forward, Forward, Low Kick
Animality ( Sweep Range ): Run, Run, Run, Block
Friendship: Low Punch, Run, Run, Low Punch
Babality: Down, Forward, Forward, Back, High Punch
Stage: Forward, Up, Up, High Kick

Sheeva:

Teleport Stomp: Down, Up
Ground Stomp: Back, Down, Back, High Kick
Fireball: Down, Forward, High Punch

Fatality ( Close ): Forward, Down, Down, Forward, Low Punch
Fatality ( Close ): Hold High Kick ( Back, Forward, Forward )
Animality ( Close ): Run, Block, Block, Block, Block
Friendship: Forward, Forward, Down, Forward, pause for half a second, High Punch 
Babality: Down, Down, Down, Back, High Kick
Stage: Down, Forward, Down, Forward, Low Punch

Shang Tsung:

Fireball: Back, Back, High Punch
Double Fireball: Back, Back, Forward, High Punch
Triple Fireball: Back, Back, Forward, Forward, High Punch
Volcanic Eruption: Forward, Back, Back, Low Kick 
Jax Morph: Forward, Forward, Down, Low Punch
Sheeva Morph: Hold Low Kick, Forward, Down, Forward, release Low Kick OR Forward, Down, Forward, Low Kick, Low Kick
Kano Morph: Back, Forward, Block
Sub- Zero Morph: Forward, Down, Forward, High Punch
Liu Kang Morph: Forward, Down, Back, Up, Forward
Sektor Morph: Down, Forward, Back, Run
Cyrax Morph: Block, Block, Block
Kung Lao Morph: Run, Run, Block, Run
Sonya Morph: Down + Run + Low Punch + Block
Kabal Morph: Low Punch, Block, High Kick 
Nightwolf Morph: Up, Up, Up
Sindel Morph: Back, Down, Back, Low Kick
Stryker Morph: Forward, Forward, Forward, High Kick

Fatality ( close ): Hold Low Punch, Down, Forward, Forward, Down, release Low Punch
Fatality ( close ): Hold Low Punch, Run, Block, Run, Block, release Low Punch
Animality ( sweep range ): Hold High Punch, Run, Run, Run, release High Punch
Friendship: Low Kick, Run, Run, Down
Babality: Run, Run, Run, Low Kick
Stage: Up, Up, Back, Low Punch

Secrets:

Secrets in Mortal Kombat 3 come in the form of Kombat Kodes, a new concept introduced into the series in the 3rd installment. In a Two player game before a match starts, there is a "Vs" screen, with 6 small boxes at the bottom. Each player can change the symbol that appears in these boxes. Player one changes the first three, while player two changes the second three. For each player, Low Punch changes the first box they have control over, Block changes the second and Low Kick changes the third. 

The shorthand way of writing the codes is listing the number of times each player presses each button. These are the symbols that appear after each button press:

0 MK Dragon (starting symbol)
1 MK Logo
2 Yin-Yang
3 Number three
4 Question Mark
5 Lightning Bolt
6 Goro
7 Raiden
8 Shao Kahn
9 Skull

The format the codes are written in are as follows:

Player 1 Low Punch, Block, LowKick - Player 2 Low Punch, Block, Low Kick

For example, let's take the first code listed below:

033-564

This means Player 1 presses Low Punch 0 times, Block 3 times and Low Kick 3 times, while Player 2 presses Low Punch 5 times, Block 6 times, and Low Kick 4 times.

Here is the complete listing of codes and what they do:

033-564 : Winner of the first round fights Shao Kahn
969-141 : Winner of the first round fights Motaro
769-342 : Winner of the first round fights Noob Saibot
205-205 : Winner of the first round fights Smoke
985-125 : Psycho Kombat (Dark kombat, Randper kombat, no Blocking)
987-123 : No Power Bars
020-020 : Blocking Disabled
100-100 : Throwing Disabled
466-466 : Unlimited Run
460-460 : Randper Kombat
688-422 : Dark Kombat
642-468 : Galaga (Game over for both players after playing)
707-000 : Player 1 Quarter Energy
000-707 : Player 2 Quarter Energy
033-000 : Player 1 Half Energy
000-033 : Player 2 Half Energy
239-494 : Sweeping Disabled
552-000 : Double Damage Player 1
000-552 : Double Damage Player 2
552-255 : Double Damage Both Players
101-000 : Healing Enabled Player 1
000-101 : Healing Enabled Player 2
303-303 : Healing Enabled Both Players
978-243 : Heavy Damage Both Players
334-000 : Half Damage Player 1
000-334 : Half Damage Player 2
334-433 : Half Damage Both Players
202-000 : Weakning Enabled Player 1
000-202 : Weakning Enabled Player 2
404-404 : Weakning Enabled Both Players
972-297 : Timer Disabled
978-243 : Short Battle
348-279 : Half Time
987-666 : Hold flippers during casino run (Text Message - No effect on gameplay)
282-282 : No fear = Ed button, Skydive, Max Countdown (Text Message - No effect on gameplay)
123-926 : There is no knowledge that is not power (Text Message - No effect on gameplay)

There is also one other secret...if you play 100 Two-Player games in a row, you get the play the original arcade version of Galaxian (cool! :D)