The Enchanter (EverQuest)
EverQuest's Enchanter is one of the most interesting and challenging classes in the game. The difference between a good Enchanter and a bad Enchanter can make or break a group in many cases. Why? Because the Enchanter is almost completely devoted to crowd control. Your job is to weave enchantments upon your foes to leave them spellbound and helpless while your friends kill their friends. If you think you're hot stuff because you can sheep like a pro, imagine keeping four or five crowd control spells on various mobs, and maintaining them all as your only job. The Enchanter's core role is pure support.
As if that wasn't cool enough, Enchanters are also the masters of illusion and mind altering spells. They receive personal illusions to disguise themselves as commonly found inanimate objects or any other race (manipulating their faction standing in the process), can cast spells to frighten or befriend their enemies, charm their foes into doing their bidding, and bolster the minds of their friends with mana restoring and intelligence boosting spells. The Enchanter is a completely non-standard fantasy staple class, is fun and challenging to play, and has a whole bag full of fun tricks to play with.
The Lore Master (Lord of The Rings Online)
The first time I read about Lord of the Rings Online's Lore Master class, it immediately reminded me of the Enchanter with its heavy crowd-control emphasis and pets, but there's so much more to it that makes it a unique and interesting class. In addition to having a bunch of crowd control and stunning tricks, the Lore Master can also throw out heals to her group, use powerful offensive pets like the Bear (something an enchanter couldn't do), and even mix it up in melee with the right gear and skill setup.
While the abilities of the Lore Master are numerous and complex, what really got me was the idea that this class wasn't at all magical. LoTRO's setting is very low-magic, as fantasy games go. You don't see fireballs being conjured out of thin air. The designers solved this problem by making the Lore Master more of a trickster/sage/alchemist, which is awesome from a role-playing perspective. I love the idea that instead of just summoning a fireball, the Lore Master uses chemistry to make it, or throws powders and dusts at her enemies. How many games really let you be an epic chemist? Not many.
The Mesmer (Guild Wars)
The Mesmer of Guild Wars is another class that has some things in common with my beloved Enchanter. They're all about illusions, mind-altering spells, and disabling their foes. What's cool about Mesmers, however, is that they're much more offense-oriented with their magic-- more of a psionicist than an Enchanter, really. There's just something inherently cool about locking your opponent down by making them believe that they have to move slowly or killing them with fear by making their nightmares seem real.
They also are uniquely talented in quickly casting their spells, interrupting others, and denying their opponents energy, making them speedy, flexible characters that pair well with other casting professions who shut down their enemies. Pretty much every Guild Wars character I make uses a Mesmer class for its main or secondary profession.
The Hunter (World of Warcraft)
While World of Warcraft's Hunters are so common these days that the proverbial Night Elf Hunter is practically a joke, they were a really original class when WoW launched. You can see influences in the class from the pet classes of EverQuest and the archer classes of various other games, but the idea of a self-sufficient, DPS archer is a pretty uncommon one in most games. It's pure genius, to the point where when WoW first came out and I was just reading the introductory material, this was the class I wanted to play. Ironically, I've never gotten one past 25.
However, the hunter is a great class with tons of cool tricks and fun things to do. The combination of having high ranged DPS (which can continue without mana), the high AC of mail armor, and a pet which is almost as strong as another character on its own makes Hunters the perfect soloing class. Speaking of pets, the pet system is probably the coolest thing about the Hunter. Not only do you get to pick your pet from all of the monsters in the world, but each monster type has their own abilities, you get to name them, and there's a whole mini-game in finding and collecting pet abilities. Add in stuff like Feign Death, traps, and tracking, and you have a class that's got more flavor and character than almost any other. I'd love to see this type of class carried over into more games, now that Blizzard has established it.
The Adventurer (Anarchy Online)
You've probably never heard of the Adventurer unless you're one of the few people who played Anarchy Online, but it's probably one of the coolest classes ever. Adventurers are the ultimate generalist-- a sort of techno-druid. Depending on how you set up your skills, adventurers can be melee fighters with dual swords or ranged fighters with dual pistols. They're the second best healers in the game (after Doctors) and get bonuses to traveling skills, making them ideal for exploration fans.
In addition to all that stuff, Adventurers have the ability to morph themselves into a variety of animal forms. Everything from tiny rodents, to sabertooth tigers, to dragons. Yes, dragons. A class that can do everything I just talked about and turn into a flying serpentine bringer of death on top of it automatically gets a slot in this column, as far as I'm concerned. The technological flavor of the Adventurer combined with the magical elements mesh together to make a ridiculously fun and versatile class.
Now, these are just my opinion of what the coolest classes in various games have been. As you can see, I lean towards generalist support roles (are you surprised that my WoW main character is a Shaman?). Your mileage may vary, and there are plenty of games I've only played in passing where I may have missed a really unique and interesting class. Honorable mentions that come to mind are the Dragons of Horizons, Masterminds in City of Villains, and Beast Tamers in Ultima Online.
Do you have any more to add to that list?