Multiclassing

Ever since the the original D&D made every Player Character pick a class, players have chaffed at the restrictions that imposed. Why can't I build half the characters in The Lord of the Rings in this system? Why does the Fighter have to only be good with his sword? Why can't the Rogue do things besides backstabbing, shooting, and trapfinding? Why is the Wizard only able to do fucking everything in the right hands with the right mindset? Whose bright idea was it to make Elf and Dwarf their own classes?

As a result, uncharacteristically following public opinion and appeal, the tabletop RPG market actually responded by giving the player the ability to multiclass, gaining many of the advantages of two different classes at once, and usually sacrificing some of the advantages both ways to maintain balance, or suffering a penalty of some kind to compensate.

AD&D

First and Second Editions offered two different ways to multiclass. Confusingly, one was called multiclassing, while the other was called dual-classing. Both were also class and racial locked, because fuck you, Gygax's Asperger's was married to frustrating and arbitrary restrictions, and he didn't care who knew it!

Multiclassing was limited to different races for different class combinations, as well as the stat, race, and alignment restrictions for both. Essentially, you had all the advantages and disadvantages of two different classes at once (a fighter/mage, for instance, couldn't cast spells in armor), but had to split all the XP earned up among all the classes you had levels in. A fighter/mage would gain half as much XP as everyone else in both classes, a fighter/mage/thief would divide his XP into thirds, a (Dark Sun-exclusive) mage/fighter/cleric/psionicist would split his XP into quarters, etc. And since classes in 2nd Ed. all leveled at different rates this quickly turned into a clusterfuck for the player to manage.

Dual-classing was human-exclusive, because of course humans have to have the potential to be the best at anything. At any point in his or her career, a human could, after gaining a level, decide to dual-class into another he or she met the stat requirements for. Once he did so, he would immediately forget how to do the things his first class taught him to do and be reset to one in his second class to keep going from there. Once she had gained enough experience in the second class to have the same level as his first, he would regain all the abilities the first class gave her. It would never advance again, but he or she would keep on leveling the second class. Thus, the usual way of doing it was to start as a muscle class (for the early advantage) before becoming a spellcaster (for the late-game scaling).

Both were, as was the rule back then, unnecessarily complicated, messy, and a pain to manage, but could be very powerful in the right hands, sacrificing the raw power of higher levels for versatility in many possible roles at once.

3rd Edition and Subsidiaries

Third Edition changed the way leveling worked, and multiclassing is much more straightforward. Whenever a character gained a flat amount of XP, the same for every class, he could choose to gain a level in any class, provided, again, that he met any restrictions that class had. A barbarian could take a level in druid to become a rage-spellcaster. A sorcerer could take a level in cleric to gain access to some nice spells and gish potential. A ranger could take a level in rogue for skill points and special trap-disarming potential. A wizard could never take a level in any class that might dilute their world-smashing uber-powers. A fighter could take two levels in fighter before immediately reclassing into something else. The only limit was your imagination, and possibly the viability of building your bizarre chimera-character, though there was one frustrating bit of unnecessary wonkery: XP penalties. Taking anything but one of your "racial favored classes" as a class beyond the first gave your character a 5% XP penalty, and it stacked for every other class you took. A lot of DMs didn't see fit to enforce that, and Pathfinder outright removed it, but it still sucked for those who did have to deal with it.

This edition also gave rise to the idea of a dip class, one that a character would take only a few levels in for front-loaded initial benefits before reclassing out of for better-scaling ones. The archetypical example is the pre-Pathfinder fighter, which offered good hit-points and feat support for the first few levels before falling off later.

In general, this class and multi-class system was pretty fun and functional. It had a lot to offer for people trying to build their own unique characters within a sane framework, and the addition of Prestige classes helped encourage otherwise-neglected combinations. However, it had... internal problems. Monte Cook's insane caster fetish ensured that magic classes were just balls-out more powerful than others, while a lack of broad quality control often meant that many classes were often poorly-designed and broken, in both directions. Redundant class powers didn't always stack, and pure casters rarely took levels outside their first when doing so meant losing a full level's worth of spells and caster-level progression.

Pathfinder changed up the formula a little with the archetype system, which lets players play modified versions of base classes, which took a lot of the necessity out of multiclassing. In fact, one of their new books is essentially a series of AD&D-style blent classes, which take half their features from two different classes! (The Brawler is a fighter/monk, the Bloodrager is a barbarian/sorcerer, etc.) While actual multiclassing has thus become more rare, the spirit of multiclassing is stronger than ever.

4th Edition

Quit your whining son. It was a thing.

Characters could take a feat at any level to multiclass in another class. Doing so was more akin to dipping or selecting an archetype than anything else: while their primary class would still be dominant, multiclassing allowed the player to select powers, skill training, and features from the off-class, and counting as it for the purposes of unlocking Paragon Paths and Epic Destinies. Only a bard could do it more than once, but it could still be fun under the right circumstances.

There were also hybrid classes that were (ironically) essentially the Pathfinder model. A hybrid picks two classes to hybridize. He or she receives only the armor proficiencies that both classes have, but gets all the weapon and implement proficiencies and trained skill choices from both classes, hit points and healing surges equal to the average of the two classes (rounded down and before adding Constitution bonuses), and their choice of traits, abilities, and powers from any one of the two classes they hybridize. It's labor-intensive, but interesting, to say the least.

5th Edition

Multiclassing in 5e is essentially a blend third and fourth edition multiclassing. Whenever you level up you can choose to take a level in another class, as per third edition rules. However, multiclassing doesn't give the same proficiencies as taking the first level in a class and the earlier class will remain dominant, like in fourth edition non-hybrid classes. For example, a Fighter 1/Bard 1 will have heavy armor, martial weapons, etc, but will only have one instrument and one skill, whereas a Bard 1/Fighter 1 will have three instrument and skill proficiencies, but no heavy armor.

Of course, multiclassing has been made somewhat less prevalent by the addition of archetypes, though unlike Pathfinder archetypes these are automatic class components rather than optional features that change the class. Each class has at least two branches (most have three, and some have as many as five) that they can take, specializing their abilities in some way. For example, a 3rd level fighter can become a Champion (better crits, better Athletics checks, eventually gains beefy fast healing), a Battlemaster (gets battle maneuvers to trip/disarm/frighten enemies, has more benefits for fighting tactically and being a party leader) or a retooled Eldritch Knight (who can cast spells and eventually learns to do it whenever making attacks). As a result, mixing two classes' worth of abilities is much easier without true multiclassing, though, as per the Pathfinder equivalent, the spirit of playing a "mixed" character is still there.

Other Systems

We can summarize this fairly easily: systems with rigid level progression generally offer a way to multiclass, systems with point-buy progression generally do not. (Systems without classes like GURPS can be safely taken off the table for obvious reasons.)

For instance, games in Warhammer 40,000 Roleplay (Dark Heresy, Rogue Trader, Only War) almost never give the player the ability to multiclass, while Exalted never lets a player play two castes at once or to change one's class once it's locked in. Legend of the Five Rings may offer the ability to mix schools within its three classes under the right circumstances, or even to take "schools" that offer abilities similar to the other classes, but bushi are still bushi, shugenja are still shugenja, and courtiers still don't get a fun Japanese name. Star Wars: Edge of the Empire comes the closest, letting players buy access to other specialization skill trees from outside their base class at a price hike, but even here that's more of a way to gain access to skill unlocks and unique abilities than to seamlessly blend strengths. In fact, "classes" in these games tend to mostly exist to grant price breaks and "unlocks" for particular skills and talents than to offer hard benefits!

In general, it's easy to see the reason why: the freeform nature of progression in these systems offers the same versatility that multiclassing normally would in others. And while in theory multiclassing lets you do anything, in practice players pick classes that offer some kind of synergy anyway, so said point-buy systems are essentially just adding guard rails to the established practice.