Quick class tl;dr

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The 5 Base Branches

Swordsmen: Tank and Melee attacker.

1.1 Knight: Very tanky, can mob efficiently due to hitlock prevention (Endure) and faster walk speed (Peco Riding). Can mob and aoe kill by itself very well, although this often requires good gear.
Damage: 3/5
Tank: 5/5
Solo: 4/5
Party: 4/5
Costs: 5/5
Guide Resources: https://web.archive.org/web/20181113082150/http://wiki.originsro.org/wiki/Knight

1.2. Crusader: Similar to the Knight with less focus on melee damage and more on support. Their main gimmick is "Devotion" (absorbing damage for others) and various defensive skills. Also can heal, and has good AoE holy dps against undead+dark mobs. Versatile, but limited by skill points.
Damage: 3/5
Tank: 4/5
Solo: 1-4/5
Party: 5/5
Costs: 4/5
Guide Resource: https://web.archive.org/web/20181113082150/http://wiki.originsro.org/wiki/Crusader

Magician: The classic magic, ranged caster.

2.1 Wizard: The quintessential "nuker". Amazing AoE dps, but very slow cast times without help. Can solo efficiently without gear, although geared classes often end up faster eventually. Highly desirable for parties.
Damage: 4/5
Tank: 1/5
Solo: 4/5
Party: 5/5
Costs: 2/5
Guide Resource: https://web.archive.org/web/20181119172042/http://wiki.originsro.org/wiki/Wizard

2.2 Sage: A support-ish mage with utility skills Has incredibly strong spells for any PvP content (such as dispelling ALL your buffs), sometimes needed for MVPing. Has the highly desirable "Endow" buff to imbue physDPS classes with elements.
Damage: 3/5
Tank: 2/5
Solo: 4/5
Party: 2/5
Costs: 2/5
Guide Resource: https://web.archive.org/web/20181119172042/http://wiki.originsro.org/wiki/Sage

Merchant: Exactly what it says on the tin. A class focused on commerce. Can open shops to sell items.

3.1 Blacksmith: The Blacksmith can create elemental weapons which are very useful for early game, undergeared characters. It has Mammonite, a strong DPS skill at the cost of zeny.
Damage: 4/5
Tank: 3/5
Solo: 2/5
Party: 2/5
Costs: 5/5
Guide Resource: https://web.archive.org/web/20181119172042/http://wiki.originsro.org/wiki/Blacksmith

3.2 Alchemist: The alchemist brews potion. It can create a Homunculus, essentially a pet minion that fights for them. The Homunculus lets you level practically "afk" in certain places
Damage: 1/5
Tank: 2/5
Solo: 1/5 (5/5 with Homunc)
Party: 1/5
Costs: 2/5
Guide Resource: https://web.archive.org/web/20181119172042/http://wiki.originsro.org/wiki/Alchemist

Thief: A melee attacker relying on dodge (flee) rather than tankiness.

4.1 Assassin: Gains stealth and more single target capabilities. Quite the unpopular class, but can level efficiently with mobbing + grimtooth.
Damage: 3/5
Tank: 3/5
Solo: 3/5
Party: 1/5
Costs: 4/5
Guide Resource: https://web.archive.org/web/20181119172042/http://wiki.originsro.org/wiki/Assassin

4.2 Rogue: Also stealthy. Can copy almost any skill in the game, which leads to some gimmick builds, for example stacking its double attack with the Monk's triple attack, using the Knight's bowling bash skill with a bow. Generally not that great outside of PvP.
Damage: 2/5
Tank: 2/5
Solo: 3/5
Party: 2/5
Costs: 4/5
Guide Resource: https://web.archive.org/web/20181119172042/http://wiki.originsro.org/wiki/Rogue

Archer: The ranged DPS. Extremely efficient throughout the entire game due to versatility and range having inherent advantage against mobs (you don't get hit).

4.1 Hunter: The ranged DPS with extra sauce. The ultimate "farmer", very budget efficient since you either get FREE elemental advantages through different arrows, or FREE added damage if you have Orc Archer Bow + Endow.
Damage: 4/5
Tank: 1/5
Solo: 5/5
Party: 2/5
Costs: 2/5
Guide Resource: https://web.archive.org/web/20181119172042/http://wiki.originsro.org/wiki/Hunter

4.2a Bard: A male support class with the remnants of its ranged DPS efficiency still lingering. The Bard plays different songs providing strong buff effects to anyone standing in their AoE. "Poem of Bragi" may be the most overpowered buff effect in any MMO, essentially nullifying skill cooldowns and cutting cast times short, which usually scales their DPS to high heavens.
Guide Resource: https://web.archive.org/web/20181119172042/http://wiki.originsro.org/wiki/Bard
4.3b Dancer: A female support class with the remnants of its ranged DPS efficiency still lingering. The Dancer's dances work the same as the Bard's songs, but provide different effects. Unfortunately, they are nowhere near as powerful as "Bragi", making Dancer less desired. Nevertheless the class IS often desired, especially for "Service for You" dance which minimizes SP costs.
Damage: 3/5
Tank: 1/5
Solo: 4/5
Party 5/5
Costs: 2/5
Guide Resource: https://web.archive.org/web/20181119173847/http://wiki.originsro.org/wiki/Dancer

Acolyte: The support class. The buff bitch. The heal slut.

4.1 Priest: The same but more. The priest is the quintessential supporter, with its buffs being extremely desired. In short, everyone will suck your dick for playing this class. As if that wasn't enough, you can also solo extremely efficiently and on a budget with the "Turn Undead" skill one-shotting undead mobs.
Damage: 1/5
Tank: 4/5
Solo: 1-4/5
Party: 5/5
Costs: 1/5
Guide Resource: https://web.archive.org/web/20181119173847/http://wiki.originsro.org/wiki/Priest

4.2 Monk: The Monk can be many things. "Asura Strike" is the most powerful single-target skill in the game by far, making them the ultimate boss killers. "Steel Body" is the most powerful tanking skill in the game, making them the ultimate boss tanks. They are excellent in PvP, "Investigate" lets them farm high def mobs, they still get access to the Acolyte's Buffs and Heals despite being a DPS class. The "Body Relocation" skill lets you essentially teleport anywhere on the screen, allowing extremely fast movement. The catch is that you can't do all of it at the same time, your skill levels are spread very thin and optimized stat builds will differ wildly per activity. You should know the goals you have with the class.
Damage: 1-5/5
Tank: 2-5/5
Solo: 1-4/5
Party: 3-4/5
Costs: 3-5/5
Guide Resource: https://web.archive.org/web/20181119173847/http://wiki.originsro.org/wiki/Monk

Tanscendence

Transcendence, also called Rebirth, is when you have 99/50 character and essentially turn into a 1/1 Novice again, except this time a High Novice. This advances your classes to their Rebirthed version which is much more powerful. They gain new skills, 20 additional skill points (job 70 max), more stat points (starting with 100 instead of 48) and have 25% higher HP & SP. Moreover the most powerful armor gear is often only equippable by trans classes.

The trans classes vary from essentially being the same but stronger, to gaining game-changing new skills. tl;dr for the changes

Knight > Lord Knight: The same but stronger
Crusader > Paladin: The same but stronger.

Wizard > High Wizard: The same but stronger.
Sage > Professor: Gains even more utility skills, which are absurdly good. For example he can swap SP with a party member, then convert his own HP to SP to get them back. Infinite SP for everyone.

Blacksmith > Whitesmith: The same but stronger.
Alchemist > Creator: This class gains an absurdly powerful single-target skill which rivals or even surpasses "Asura Strike" in boss killing potential. Somewhat expensive but very, very strong for bossing.

Assassin > Assassin Cross: The same but MUCH stronger and no longer a joke class. Gains a poison skill that quadruples your ATK, and a range skill that can be quite good DPS against certain enemy types
Rogue > Stalker: His copied skills can no longer be overriden. Uhhh that's it really.

Hunter > Sniper: The same but stronger.
Bard/Dancer > Clown/Gypsy: Not much of a change.

Priest > High Priest: Gains another powerful buff that reduces all incoming damage by half. It's exactly as good as it sounds.
Monk > Champion: The same but stronger.

Expanded Classes

These classes were added as extras beyond the basic 5. The trade-off is that they cannot Rebirth, which in most situations means they are eventually outpaced, especially since they don't have access to the best gear (which tends to be rebirth-only). Generally speaking these classes fall between pre-trans and trans classes in terms of strength.

Taekwon Boy
A class that kicks without a weapon. It has access to every single element to attack with. Their base class is fairly useless, however if they are in the top 10 mission rankings (hence they are called "Taekwon Rankers") they gain powerful buffs, including tripled HP and infinite kick combos. A ranker has the advantage of being effective with very little gear. Overall they are a niche and gimmicky class, however.
1.1 Taekwon Master / Star Gladiator
This class is bizarrely complicated. It can select buffs for specific maps, specific mob sizes and days of the week to increase damage. It can employ the "Heat" skill as effective DPS against boss mobs as they have no push-back and will continually get hit by the Heat AoE. Book attacks + Kicks can be also be viable with enough investment. Basically it can be both a good boss killer and exp farmer, but will be cucked into specific niches and requires planning, foresight and friends.
1.2 Soul Linker
Mainly a buff bitch whose buffs (soul links) can be EXTREMELY strong depending on the choice. Every link is different, some are always useful, some are situationally strong, some are memes. It is better to look at them in detail. They also have a somewhat viable single target DPS skill, but generally not used as a farming class.
Ninja
Can be both a physical and magical DPS. Actually quite strong single target either way, and a viable early farming class, their main limiting factor is simply the lack of rebirth.
Gunslinger
A ranged physical DPS like the Hunter, with access to different types of guns that offer different skills. Again, its DPS is quite strong and it's a viable farming class whose effectiveness is only limited by the lack of rebirth.