Well, here's a quick rundown of the classes.
1st classes: You can promote to these after lv10.
Swordsman - A class which wields swords and aims at dishing out a lot of damage with basic attacks. If you're a lance, you can promote to Crusader and if you're Carsise, you can promote to Champion.
Herbalist - A class which can either go a bit more supportive, aiming at maxing out Buffs, while maxing their damaging spell "Spiritual Bolt" will result in a more battle-focused mage with decent support from the Heal spell. Only Phyllis and Ami can become herbalists, and this class can either promote to Seal Master or Cleric (Regardless of Race)
Explorer - A magic damage class, not much to say about this one (Its 2nd promotion has more information). Lance, Phyllis and Ami can be an Explorer, later promoting into Voyager.
Hunter - The only ranged class which aims at dealing physical damage. It uses Bows and has a variety of bow spells. You can later switch to gun by promoting to Sharpshooter.
2nd classes: You can promote to these after lv40.
Crusader - If you're a Lance swordsman, this is the class you'll promote to. It's an assassin class with a stealth mechanic. It aims at dealing out a lot of damage with its stun and swift attacks. If that's not enough, "Illusion Slash" can finish your opponent off, or, if you're strong, outright kill them in one hit!
Champion - If you're a Carsise swordsman, you'll promote to a champion instead. They decided not to use dual swords, and instead focus on wielding a giant greatsword, axe, hammer or other big weapons. They discard some damage and swiftness from crusaders, and instead aim at tanking a lot, while dealing out decent damage.
Seal Master - If you're a Herbalist, this is one of the classes you can promote to. As its name would imply, it excels in Sealing their targets, rendering any target which falls into the SM's clutches useless, while still providing a little bit of support through heals and buffs.
Cleric - If you're a Herbalist, this is the other class you can promote to. Clerics are more supportive classes as they have a wider variety of buffs and their heals are much stronger. Moreover, if you decide to go for a battle cleric, you'll dish out decent damage while being protected with your SP shield.
Voyager - If you're an Explorer, this is the class you'll promote to. Not only does it have an AoE magic spell which is excellent for leveling, but it also possesses skills which you can use at Sea. It can either go for a tanky build, dealing little damage but surviving quite a beating, or go for a more damage-oriented build, but being incredibly frail.
Sharpshooter - If you're a Hunter, this is the class you'll promote to. It generally drops the bow and starts using a gun. The gun deals more damage, but has less maneuverability and speed than a bow does. It also has seals, not as powerful as the seal master's, but it can still render your target nearly useless. If you've really liked the Hunter playstyle, you can still play this class with a bow, although there is very few strong high level bows in the game to make it work.
Hope this was in some way useful.