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. 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 get all the exclusive special rules to themselves. 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. 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.

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 bards could do it more than once, but it could still be fun under the right circumstances.

5th Edition

...I don't actually know enough to write this section. Bail a brother out if you do.

Other Systems

We can summarize this fairly easily: systems with progression generally offer a way to do this, systems with point-buy generally do not. (Systems without classes 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. In fact, "classes" in these games tend to mostly exist to "unlock" price breaks for particular skills and talents!

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.