Fighter, Wizard, Thief, Priest. These, of course, represent the "core" character classes found across the spectrum of many FRPGs, and has been a staple since very early on in the hobby. Not everyone agrees with these choices, of course. James Maliszewski is famed far and wide for his low opinion of the thief class, and I myself have questioned the inclusion of the cleric/priest as an archetype. Some games add more, some few trim down the list, still others expand them greatly through sub-classes, prestige classes, kits, etc.
There is another school of thought that says that classes as a concept are flawed and limiting. These are the designers who give us systems built on skills, or skill bundles. By careful choice, using such rules, one is able to come up with the precise "character concept" that one desires, whether it be a demon-summoning swordsman or a wizard who's a gourmet chef and wields a pair of pistol-crossbows. Some systems attempt to split the baby by having both classes and skills.
I make no claim as to one method being more "old school" than the other; the one was used by A/D&D for many years, and the other was used at least as early as the first edition of RuneQuest, and I for one would be the last person to question the old-school "chops" of RQ. I think it ultimately boils down to personal preference.
As the skill system was introduced as a reaction to the class system (by the nature of the timeline of development of RPGs), it stands to reason to discuss why character classes were considered the standard in the first place. Coming from the world of miniatures wargaming, a "wizard" or "hero" was just another unit type. A single figure represented only a single individual, but that individual was worth the same as a squad of normal soldiers, or had special abilities akin to a catapult or cannon. An "arch-mage" or "super-hero" was merely one such special unit writ large (thence the notion of level advancement, but I anticipate myself).
When the Great Leap Forward came, and players took on those single characters as their own, it made sense that they would not want to play one of the "grunts", but rather one of the special types. This makes sense on a practical level as well; stick a sword in a grunt, and he dies. Stick a sword in a super-hero-type, and he has at least a chance of surviving. Long-term character survival makes for more enjoyable games, as it reduces the need to play a new character every one or two games just because an orc rolled a hit with an arrow.
Not wanting to get all Joe Campbell-esque, I should point out that three of the four archetypes named at the beginning of this post are to be found in Tolkien. They cover the three bases by which most games describe the abilities of any character; physicality, mentality, and nimbleness, thus covering in a broad sense most of the threats they are likely to encounter in the game. (These are, by design and necessity, generalizations of course.) I am halfway tempted to add bard as an archetype, and pursuasiveness as a fundamental base, but I resist for a variety of reasons, chief of which will be touched on below.
In a very real sense, character classes represent a intentional break on the creativity and boundary-pushing of the player, allowing the designer to build in implicit limitations and assumptions about the campaign setting by limiting such choices. In a campaign loosely based on medieval Europe, for instance, allowing players to choose to play a Japanese Ninja or an Aztec Jaguar Warrior might prove problematical. (Of course some settings are deliberately "wild and woolly" in such regards, but that's a little beyond the scope of this post
A skill-based system, on the other hand, is a bit harder for the designer to "reign in" the creative impulses of the player. Much more scrutiny is required on the part of the game master, who is placed in the position of at least having a cursory examination of all the characters' chosen skills so that nothing potentially setting-bending is within. And this need not be intentional on the part of the player, or even an example of min/maxing; it's impossible to fully explain all of the implications of any setting at the time of character creation, thus leading to inadvertently inappropriate choices of skills. For example, if I have a vaguely Polynesian-based setting where short island-hopping is the norm, a character taking Oceanic Navigation is going to throw a monkey-wrench into the works if long cross-ocean trips are the purview of chiefs and the sons of chiefs and I didn't happen to mention it (or did, and the player took the skill anyway).
The concept of skill-bundles is an interesting one, and represents a sort of cross between the two concepts. That is, skills aren't taken a la carte, but rather in, well, bundles of related skills. A warrior-type might take a skill bundle including swordsmanship, archery, forced marching, and mounted combat. A wizard-type might take a skill bundle including ancient languages, basic alchemy, and thaumaturgy. And so forth. That gives some control to the game designer, who can craft the skill bundles into appropriate groupings, while giving some more flexibility to the players, who could choose several skill bundles to create their character
I confess I find the concept of specific skills limiting in another way, however. By defining every single skill a character possesses, everything that is not permitted becomes forbidden. For example, I might have a thief character who grew up in a lakefront community. But unless I specifically take the swimming skill, I can't swim. Compare this to the situation in a class-based game, where that same character might either a) reasonably be assumed to be able to swim based on his background or, b) at least be given the chance to have such knowledge by the game master. But where swimming is a skill that must be deliberately chosen, it can't just be assumed.
There is another qualm I have about over-statisticating* character abilities. That is the elimination or restriction of actual player actions. This has been a tension inherent in the very concept of "charisma" as an attribute in A/D&D since the beginning. If my character has a charisma of 17, and I'm trying to convince the bandit chief to let me and my party go, how does the game master adjudicate my success? Do I just roll a die based on my statistic? Do we role-play out the situation, and if I as a player am convincing, we go free? Is it perhaps a mix of the two, with the game master either rolling with a modifier based on my performance, or being more lenient with my hemming and hawing because my character has a high charisma? I use charisma as an example, but the principle applies to skills as well as statistics. If I, as a player, know enough about sailing to build a raft, should my character be denied that same knowledge? Is that meta-gaming, or common knowledge? In a game with skills, it's meta-gaming. In a game without them, where secondary knowledge is either assumed or left to common sense, it becomes more murky.
* I'm quite sure that's not a real word. Well, it is now.