Different classes portray what kind of combat experience you a) have, and more importantly b) use. Here is a complete list.

Singles (one class set.)
Swordsmaster: Uses swords as a dominant weapon. Staunch defenders with high offense potential.
Brawler: Uses hammers and axes. High damage attacker with limited defense, usually extra tough.
Templar: Uses pole arms and Spears. High defense, but lacking in offense potential.
Assassin: Peak offense, but abysmal defense. Hits to kill, uses trowing weapons and close quarters weapons.
Paladin: Peak defense, offense is awful. Uses a shield as a weapon usually. Often has a sword as well. Tries to have to opponents weaken themselves.

Ranger: Offense type Archer. Uses bows and crossbows for attack.
Archer: Defense type. Uses bows and crossbows as suppression fire.
Gunslinger: Uses guns. Balanced class with high offense and defense potential.
Bomber: Uses explosives as an attack. No real defense, but incredibly high offense.
Artillery: Uses explosives defensively. Can't often attack with choice weapons, so holds the line instead.

Mage: Predominantly uses magic. Balanced offense and defense, with the right spells. Often physically weak.

Doubles (2 classes.)
Spellsword: Mage and Swordsmaster.
Dreadnaught: Brawler and Paladin.
Executioner: Gunslinger and Assassin.
Destroyer: Gunslinger and Brawler.
Support: Archer and Templar.
Spellbow: Mage and Ranger.
Reaper: Gunslinger and Swordsmaster.
Berserker: Bomber and Brawler

Omniclass: A type of its own. When someone has more than 2 classes, or the classes they have aren't a pair set, they are called Omniclass. In Axel's case, he is Swordsmaster, Brawler, Gunslinger, Templar, and Assassin.

Classes can change over time. For example, let's say Callie decides she likes swords. Well, she would become a Swordsmaster. Easy enough. And an excuse for leaving out spoilers.

Sorry this part is small. There isn't anything else that fits with this topic.