In classic Diablo, there are three basic monster types: demon, animal, and undead. There is no way to tell which type a particular monster is from within the game; it's left to the user to guess. Most players aren't even aware that these monster types exist, as the game's manual makes only the vaguest allusions to them. In spite of this, monster types can have a important effect on gameplay.
First and foremost, the type of monster you are attacking modifies the amount of damage you do with different weapons:
Weapon Type | Undead | Animal | Demon
Sword | ×0.5 | ×1.5 | ×1.0
Club | ×1.5 | ×0.5 | ×1.0
Axe/Bow/Staff/Hands/Feet | ×1.0 | ×1.0 | ×1.0
This is supposedly to reflect the idea that a bashing weapon (like a club) would be more effective against a skeleton than, say, a crossbow, and that a thrust sword would be more lethal to a standard animal than, for example, a swung axe. This makes sense, in a weird sort of way.
Monster type is also important when using the spell Holy Bolt, as it is only effective against the undead. The sole exception to this rule is Diablo himself, who takes damage in spite of being a demon.
The only other effect monster type has is on Civerb's Cudgel, which has the attribute +200% damage to demons, but does normal damage to both other types.
There are 22 different monsters total in Diablo, each of which has 4 palette swapped variations. Here's the breakdown, including monster type and dungeon levels on which they appear:
Monster | Type | Dlvls
Zombie | undead | 1-5
Skeleton | undead | 1-5
Fallen One (spear) | animal | 1-5
Fallen One (sword) | animal | 1-5
Skeleton Captain | undead | 1-6
Scavenger | animal | 1-6
Skeleton Archer | undead | 2-6
Winged Fiend | animal | 2-6
*Familiar | demon | 6-8
The Hidden | demon | 2-10
Goat Man | demon | 4-9
Goat Man Archer | demon | 4-9
Overlord | demon | 5-12
Gargoyle | demon | 5-12
Spitting Terror | animal | 6-14
Horned Demon | animal | 7-12
Magma Demon | demon | 9-11
Lightning Demon | demon | 9-14
Balrog | demon | 10-15
Viper | demon | 11-15
Succubus | demon | 12-15
Knight | demon | 12-16
Mage | demon | 13-16
*Familiars are one of the four variations of Winged Fiends; this is the only occurrance of monsters with the same graphic having different types.
There are three special boss monsters in Diablo: The Butcher, Skeleton King, and Diablo himself. They appear on dungeon levels 2, 3, and 16 respectively. The Butcher and Diablo are demons; the Skeleton King is undead.
The earlier stages of the game are loaded with undead and animals, but the undead disappear completely after dungeon level 6, and animals become relatively rare. The second half of the game is overwhelmingly dominated by demons.
What have we learned today? Swords are generally superior to clubs in the late game, which is when it really matters. Holy bolt is damn near worthless, except against the Skeleton King. And, of course, Civerb's Cudgel kicks ass. | 
