Yes, there are characters. This is still in the eight-bit world of games, where characters don't get much characterization, but there are heroes.
Ortega's HeirThe son (or daughter) of the great hero Ortega. Shortly after you were born, your father departed on a quest to defeat the fiend Baramos, never to return. In his absence, it falls to you to take up his quest and save the world. A Magic Knight and all-around fighter with access to powerful attacks and spells. Good thing the hero is so well-rounded, since you cannot change your own class.
Tropes associated with the Hero:
The Soldier/WarriorSoldiers are your obvious choice for defense. They are able to use the strongest and most durable weapons and armor in the game, and have the most hit points of any class, but are slow as molasses.
Tropes associated with the Soldier/Warrior:
The Pilgrim/Priest/ClericYour designated healer. Unlike many games, healers in Dragon Quest can become decent attackers and can learn some wind-based magic and other spells.
Tropes associated with the Pilgrim/Priest:
The Wizard/MageYour designated combat mage. Very weak physically, but are capable of blasting enemies into little piles of dust with their magic, as well as throwing out combat buffs. Always put in the back.
Tropes associated with the Wizard/Mage:
The FighterThe physical Glass Cannon. Doesn't use much equipment. In fact, doesn't really need equipment. A poor man's alternative to the Soldier (though sacrificing nothing in sheer power), but if you prefer two physical and one magic, you will have a Fighter.
Tropes associated with the Fighter:
The Merchant/DealerThe only one of the characters you need to have to complete the game, the Merchant is the Jack of All Stats besides your healers, only not magical. Though they might not have the punch of a Soldier or Fighter, they have the ability to appraise items and find extra gold after battles.
Tropes associated with the Merchant/Dealer:
The Goof-off or JesterWhat are these guys doing on the battlefield? The higher their level, the more they goof off when they're supposed to be fighting! And even when they do what they're supposed to, all their stats except luck are mediocre at best. Not the most reliable of companions... At level 20, however, they can become a Sage without needing the Book of Satori.
Tropes associated with the Goof-off/Jester:
The SageA mysterious and rare profession, Sages learn all the spells of both Priests and Mages. Nobody is capable of simply starting as one, however; it takes special training and the rare and highly-valued Book of Satori...
Tropes associated with the Sage:
The ThiefA Fragile Speedster that only appears in the Remakes. While the player cannot manually command them to steal, they have a chance of automatically swiping items from their defeated opponents... and the higher they level, the better their chances.
Tropes associated with the Thief:
Friends and Allies
OrtegaThe Hero's father. Fought a monster over a volcano and fell in.
Tropes associated with Ortega:
The Hero's MotherExactly What It Says on the Tin. Has no canon name.
Tropes associated with The Hero's mother:
DivinegonA god who grants you wishes if you defeat him.
Tropes associated with the Divinegon:
KandarA thief whom the hero crosses paths with several times. A vicious, axe-wielding brute.
Tropes associated with Kandar:
OrochiA monster on the island of Zipangu who requires sacrifices.
Tropes associated with Orochi:
Boss TrollA vicious beast who plagues the realm of Samanao by pretending to be the king.
Tropes associated with Boss Troll:
BaramosThy enemy shall be the archfiend Baramos.
Tropes associated with Baramos:
King HydraA palette swap of Orochi. Kills The Hero's father.
Tropes associated with King Hydra:
BaraBomusAn evil wizard working for Zoma.
Tropes associated with BaraBomus:
BaraGonusA skeletal dragon also associated with Zoma.
Tropes associated with BaraGonus:
ZomaThe real cause of all this mess.
Tropes associated with Zoma: