When it comes to D&D-type games I am quite fond of "conservative racial stereotypes" in the style of the original rules: Certain races are better (or worse) at doing certain things, period. Issues of "game balance" aside, the answer to the old "Which races can pick which classes?" question is one of the central player-facing rules that allows me to make clear statements about the world, statements the players will care about a lot more than when I rattle off 20 minutes of flavor text.
Instead of discussing this entirely in the abstract, let's start by looking at the table of race/class combinations for my world:
The very first statement I make is that there are only 4 races and 4 classes that can be played which establishes the "outer limits" of the world (from a player perspective). The next statement is that humans are good at everything: the Us mean that humans can advance to any level in all available classes. The remaining U entries state what the various demi-humans are good at: dwarven fighters, elven wizards, and halfling rogues are all on par with their human counterparts. In fact, given their greater life-spans, demi-humans actually have a chance of exceeding human capabilities in those particular classes (see below for how this plays out).
The remaining nine entries are where things usually get controversial: the 4s and 8s mean that the race in question is limited to level 4 or 8 in a given class, the dash means that the race cannot have characters of that class at all. These are the cases that (in my humble opinion) require justification, and it's those justifications that convey most of the flavor about who these races really are in this particular world.
Looking at my table you may already "get" the flavor I am going for to some extent, but just in case I also provide detailed justifications below. Before we get to that, however, I'd like to briefly summarize how my race/class table differs from various "official" versions of D&D:
In comparison to (A)D&D 3e and its free-wheeling attitudes my table is blatantly restrictive. If you like "everything goes" just fine then you should definitely ignore this post.
In comparison to "classic" B/X or BECMI D&D my table is very flexible: Yes you can be an elven rogue or a dwarven cleric! But if you like "race classes" just fine you should probably also ignore this post.
The comparison to OD&D is more complex: Dwarven fighters are just as good as human fighters (instead of peaking at level 6 as in OD&D), elven wizards are just as good as human wizards (instead of peaking at level 8), but both dwarves and elves are restricted as rogues (thieves) and there are no elven clerics at all (which is more restrictive than OD&D on both counts). The elven and halfling fighters are unchanged.
The comparison to AD&D 1e is even more convoluted: Gary allows non-player clerics for dwarves and elves (but curiously not for halflings); I'd rather have a universal reason instead of treating PCs and NPCs differently. Gary stops dwarves, elves, and halflings at levels 9, 7, and 6 as fighters; he clearly gives a nod to dwarves being better at killing things, but his distinction is too muddled for my tastes. Gary does let elves advance to level 11 as wizards; as far as campaigns go, this is actually pretty close to my thinking (see below). Finally, Gary gives all races unlimited advancement as rogues, just like OD&D does; that I have a hard time with, particularly in the case of dwarves and their sausage fingers.
Does my table seem too random for your taste? Personally I find OD&D and AD&D 1e even more random, especially since those rules give very little in terms of justification or explanation. So in an effort to rectify that let me try to run you through my thinking on a class-by-class basis:
Clerics are essentially religious fanatics, an attitude that I see neither in elves (too "scientific" or "arcane" or "enlightened" as it were) nor halflings (too "laid-back" for sure, they'd at most be fanatics about food and wine). Dwarves and humans on the other hand can clearly be block-headed enough to spout religious dogma for a long time; dwarves just eventually calm down and start to realize that there's more to life than that. Humans on the other hand rarely develop far enough (not in terms of levels mind you, but in terms of overall enlightenment about the world and their place in it).
Fighters either kill things or organize others to do the same on a large scale. Every race has some potential for slaughter, but only dwarves and humans are dedicated and aggressive enough to perfect their techniques: Dwarves to defend their mountain homes (and the rest of the world) from the endless hordes of the underdark, humans almost invariably to accumulate wealth and power. Elves tend to ignore the petty squabbles of the short-lived races. They have plenty of time to "wait things out" in the relative safety of their ancient forests and most of them don't see the point of "playing soldier" and risking their (almost) immortal skin as it were. Halflings are neither fond of the discipline required nor the decided lack of comfort military ventures bring with them. In the end they also don't much care for a "fair" fight and would rather overwhelm those "ugly biggens" who want to steal their pipe weed with guerrilla tactics.
Rogues are fundamentally opportunists, they only differ regarding the kind of opportunity they prefer to exploit. Dwarves don't have much patience with this approach to life: You're either doing it right or you're doing it wrong, and if you're doing it right it's honorable and you don't have to sneak around. Elves do enjoy the mischievous aspects of life as a rogue, but they also eventually tire of the "inelegant" physical approach when magical techniques would simply guarantee success. Halflings however couldn't feel more at home with the idea of tricking others or overcoming a wide variety of obstacles with hairy toes and nimble fingers. It's pretty much the only thing exciting enough to keep them away from home during pie season.
Wizards are "academic technocrats" who unravel the mysteries of the multiverse in order to bend it to their will. Well, at least the human ones do, the elven ones look at it more as "communing with the multiverse" but it all works out to the same. Either way, studying musty tomes for hours on end (without snacks even!) doesn't sit well with halflings at all. Dwarves, while not necessarily turned off by the studying, prefer the simpler truths that tradition and religion provide over having to make up their own minds about what is what. For both halflings and dwarves their natural resistance to magic also gets in the way a bit.
Of course these are generalizations, but they are "true enough" in my world to be made into rules. I find that if I don't clearly state that "dwarven wizards" and the like simply don't exist, sooner or later a player will try to roll one up. And if I say at that point that they can't, much needless sadness results.
Note that I don't rule out the existence of "priests" as opposed to "clerics" for any race. It's just that "priests" (religious figures more akin to studious wizards than bloodthirsty fighters) are not a playable class in my world. At least not yet.
I should also mention that I really only play up to level 12 or so; in practice the "soft limit" for player characters is probably closer to 10. This means that while the level 4 limit is a significant restriction, the level 8 limit is much less of one. Combined with the fact that demi-humans can have two (or three in the case of elves) classes, this means that no race has a "hard limit" of any sort to their combined levels unless they pick unfortunate combinations. But even elven fighter/rogues and dwarven cleric/rogues can make it to 12 combined levels, so that's mostly a theoretical concern.
The limit to about 12 levels ties in with my preferred campaign style as well as my choice of level progression.