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.
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, The Kombat Tomb and The Pit 3 stages.
Pit Fatality for all characters:         Just a simple Uppercut is enough.
All other stages that contain stage fatalities need the specific character's  STAGE FATALITY.

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!
           
Mortal Kombat Trilogy introduced the Aggressor, which is a bar at the bottom of the screen that gradually fills up to spell the word "Aggressor" with each hit that you inflict on your opponent. It will fill up faster if your opponent blocks. Once the meter is full, your character will be able to inflict more damage for a short period of time.

Differences between the Playstation/PC and Nintendo 64 versions: