For starters, this ranger has leveled all twelve classes to 20 and done a fair amount of research, so I can at least speak to the basics of each and what may differentiate them from other game's implementations. This is not intended to be an exhaustive class guide, nor advice on what to take for end-game (since I'm not there yet), but rather a primer that may assist newbs in narrowing the field and a collection of the info that I used in making my own class choice.
With that disclaimer out of the way, let's see if we can't grok Age of Conan's class system. The big twelve are broken down into the following sub-categories: Rogues, Soldiers, Priests, and Mages. Some classes are race-restricted (i.e. you can't roll a Stygian Bear Shaman), so you may find your choices narrowed a bit if you have a particular affinity for Aquilonians, Cimmerians, or Stygians.
Rage of Mages
Mages are perhaps the least noteworthy of Age of Conan's classes. Not because they're not fun, but because by and large they don't make use of the newfangled directional combat system to the extent that melee types do. If you're a caster, you basically spam ranged spells like every other MMORPG, though you generally look better doing it in Age of Conan. The exception to this is the hybrid Herald of Xotli, but the Demonologist and Necromancer strike me as very similar to their caster and pet-handling counterparts in many other games.
Herald of Xotli
The Herald is an interesting class, so much so that I defy anyone to find me another MMO that features a cloth wearing greatsword wielding caster class possessed of potent dark magic plus the ability to change into a demon. Still looking? Yeah, go ahead and give it up. From what I can tell at the early stages, Heralds are one of those 'easy to learn/hard to master' classes. In the right hands, the class can be deadly, just don't expect to survive if your opponents decide to focus fire.
Most one versus one encounters end up with my opponent dead or running away.
Demos are the epitome of the AoE-based caster. If you want to hurl fireballs, summon half-naked succubus pets, dole out a generous amount of crowd control, and generally make life miserable for groups of mobs, this is your class. I've also done a bit of PvP with my 21 demo, both in mini-games and in the White Sands/Underhalls portions of Tortage. Most one versus one encounters end up with my opponent dead or running away. I assure you this is not because I'm a skilled PvPer.
To start, I have to say that Age of Conan's necromancer pets are the single grossest thing I've ever seen in an MMO. They are so nasty that I used to be amazed that people would actually want to play this class. Then I rolled one, got into a couple of White Sands fisticuffs, and instantly saw the appeal. Imagine a class that does massive damage, hefty DoTs, and can do both while kiting, and you'll begin to see why Necro victims commonly shout a variation on "WTF overpowe......ugh" before face-planting on the rez pad.
Priests of the Temple
Healing is one of the areas where Funcom has thrown out the rulebook with regard to past MMORPGs. In Age of Conan, you rarely have to stare at your party's health bars since most of the healing spells are AoE-based and affect your entire party. This frees up the priest to do damage in addition to medic duties, though some classes are currently more effective at this than others.
Priest of Mitra
Community opinions are pretty divided about the usefulness of PoMs after a few notable ner....er, changes made to crowd control range. My own PoM experience has been unremarkable, primarily because I've largely played solo to this point. Prior to level 25 the class fares well in PvE with an effective knockback, but PvP has been an exercise in frustration due to long casting times and a fair amount of squishiness.
Tempest of Set
The ToS is one of those classes that you love or hate, depending on whether you rolled one or are playing against one. In keeping with Age of Conan's unique approach to the healer archetype, the ToS is one of the game's premier AoE damage dealers. The class is as adept at pumping out DPS as it is at repairing it. While many have cried for nerfs to the Tempest, the reality is that they're quite fragile. PvE through level 25 is generally a breeze, provided you can keep from aggroing the entire zone with your destructive awesomeness.
Bear Shamans, like Tempests of Set, are healer/DPS hybrids. The main difference is that the Shaman does single target damage as opposed to area effects. Shamans are also a bit heartier, given that they can equip medium armor and man the front lines more effectively than squishies wearing cloth. Looking at the feat tree, you can opt to spec yourself into a very effective healer or a very effective damage dealer.