I recommend the Fighter when you play this game for the first time. He is the most efficient when it comes to offense. His sword allows him to attack enemies farther than the Cleric and Dwarf can. His inability to use magic makes managing his inventory the easiest. He somersaults when his inventory is empty.
The Cleric is a priest who has taken up some of the ways of a fighter. Their vows prohibit them from using edged weapons so he uses a mace. Unfortunately, this mace gives him a shorter attack distance, which can be somewhat frustrating during battle. He can cast spells such as Turn Undead (kills undead), Hold Person, Striking (increases strength), Continual Light (blinds enemy), Sticks to Snakes (makes snakes that attack the enemy), and Cure Serious Wounds (heals).
The official chick of Tower of Doom! The Elf is like Final Fantasy's Red Mage, they're good at fighting and using magic. Her sword allows you to attack at the same distance the Fighter can. She can cast spells such as Magic Missile, Fireball, Invisibility, Lightning Bolt, Polymorth Other, Cloud Kill, and Ice Storm. Quite nifty, indeed!
Like the Cleric, the Dwarf's ax has a short attack distance, which can be somewhat frustrating during battle. However, he can swing his axe faster than anyone else. He performs his first three attack moves at almost double the speed! When his inventory is empty, he executes a twirling ax attack, this attack can hit twice in the air and then once again on the ground.