I'm interested in how the multiclassing rules evolved/changed throughout the history of the game. A small summary of differences should be sufficient. Please include Pathfinder in the answer. I believe it's related enough to add it.
Multiclassing was not supported initially (three brown books) unless one played an elf. The elf version of “multiclassing” was to advance as a magic-user (up to level 8) and a fighting man (up to level 4) by default. The Greyhawk supplement provided for multiclassing into the thief class for some non-humans (elves, dwarves). It also introduced half-elves who could advance in multiple classes: experience was split among classes whenever awarded. The elf-as-thief could be either a straight thief, or was triple classed (fighting man, magic user, thief). As a whole, the first forays into being something other than a single class were quite needlessly complex. The Moldvay/Mentzer BX/BECMI D&D editions dispensed with them entirely.
AD&D (1e & 2e)
Two different forms of multiclassing were available.
“Dual-classing” was only available to humans, and allowed you to start over, from 1st, in a second class, retaining nothing from the first. When you again reached the level you had before, you then regained everything from the first class, and then from then on advanced the second class (only). Paladins and monks could not multiclass.
“Multiclassing” involved splitting your XP evenly between two, three, or in one case, four different classes. Since classes in this edition required different amounts of XP to level up, your levels in each class would end up different even though the XP totals were the same. The exact combinations you could use depended on your race, alignment, the phases of the moon, etc.
D&D 3e and 3.5e
Multiclassing was a rather large part of the third edition, and in addition to being a common choice for characters, numerous “multiclass adjacent” options were written.
Multiclassing was simply a matter of choosing a different class when you leveled up, instead of whatever other class you had previously been advancing, gaining the 1st-level abilities of that class instead of whatever level you would have gained. You could choose differently again the next level, picking up the first class where you left off, advancing the second, or starting a third.
As special exceptions, you could not return to the monk class after you’d left it, and you could not take a level of paladin if you ever had a level of anything else. In my experience, these rules were rarely enforced.
Your HD at 1st level was maximized, and your skill points quadrupled. These benefits applied only to the class you took at 1st level, and not to any other class you took later.
Skills in general were complicated by multiclassing. Each skill has a maximum skill rank based on your level and whether or not it was a class skill, and buying ranks with skill points also cost more for cross-class skills. When you had multiple classes, this got difficult—the max was based on whether the skill was in-class for all of your classes, but the cost was based on whether the shop was in-class for the particular level you were taking.
Finally, if your levels in different classes were too far apart, you took fairly large penalties to the XP you gained, probably as a nod towards the XP splitting of 2e multiclassing. Each race got a favored class; your levels in your favored class didn't count to determine XP penalties—probably a nod towards 2e’s heavy use of racial locks on different multiclass options. These rules were even less frequently used than the monk and paladin restrictions.
Prestige classes, especially theurges
In addition to the base classes you could multiclass into, 3e had “prestige classes,” which you could not start as, and had to multiclass to get to, after you had met their prerequisites by leveling in some other class. Prestige classes did not contribute towards XP penalties.
For instance, the assassin class required several ranks in various stealth skills before you could take a level in it—you needed at least 5 levels in something else before you could qualify. So most assassins were actually 5th-level rogues before they became assassins.
Notably, many prestige classes, known as “theurges” after the core mystic theurge, advanced two classes’ features, enabling a way to use two or more classes without being quite so far behind, as simply taking half your levels in each tended to result in a character who wasn’t good enough with either. For example, the aforementioned mystic theurge advanced both arcane and divine spellcasting, so a cleric/wizard/mystic theurge was closer in power to a single-classed cleric or wizard, and far better than just trying to go with a cleric/wizard. Prestige classes that advanced some combination of classes became very common in this system.
Variant classes, alternate class features, substitution levels, etc.
And then on top of that, there were numerous experiments with modifying classes: variant classes, alternate class features, racial substitution levels, and so on. Many times, these variations on a class made the class more like another class, enabling a kind of “single-classed multiclassing,” because you traded some of your typical class features for the features of another class. For example, the thug fighter got better skills and could get some sneak attack instead of heavy armor proficiency and some bonus feats, making them somewhere halfway between the regular fighter and the rogue.
Multiclassing was achieved simply by taking the appropriate feat, rather than anything you did with your levels. The multiclass feat corresponded to some class other than your own, and gave you a watered-down version of that class’s signature ability. Once you took the feat, you were also entitled to take some of that class’s powers as you leveled up. Most characters could only take one multiclass feat, and most would want to.
4e included “paragon paths,” which were similar to the prestige classes in 3e, except that every character got one starting at 11th level, in addition to their heroic class, rather than having to multiclass into the paragon path. However, there was also something called paragon multiclassing, which involved basically having a second heroic class instead of a paragon path from 11th to 20th. This didn’t work very well and was rarely used.
Finally, there was also a system called “hybrid.” They printed a “hybrid” version of every class, which was literally the class cut in half, and then you could combine two of these half-classes to be your class, choosing roughly equal numbers of powers from each. In a select few cases, this was very potent; in most cases, this was one of the few ways to actually produce a crippled 4e character.
A spin-off of 3.5e, largely written in reaction to 4e, Pathfinder’s multiclassing works exactly the same as 3e except that there were no XP penalties, and favored class did something else entirely. Also, since skill points were no longer quadrupled at 1st, and cross-class skills no longer cost more or had a different max, that no longer was a factor in multiclassing.
Pathfinder also continued to have prestige classes, though most were massively nerfed and consequently rarely used. Theurge prestige classes became far less common, and there weren’t nearly as many combinations supported.
On the other hand, Pathfinder played up the variant class angle to the hilt, calling them “archetypes.” Archetypes are a huge proportion of the overall Pathfinder content, and there are far more options for mixing in some features from another class through an archetype than there are for combining two classes with multiclassing or a theurge prestige class.
Going even further, Pathfinder also tried out a number of “hybrid” classes, which were classes written to be mixes of two “parent classes.” For example, the bloodrager class was a hybrid of barbarian and sorcerer. Unlike 4e, where hybrids were a whole “system” and every class was split in half, allowing players to mix-and-match as they liked, every Pathfinder hybrid class was written specifically for that combination, and you could only “hybridize” those class combinations they actually wrote a hybrid class for.
Another approach to “single-classed multiclassing,” again rather similar to a 4e idea, this was multiclassing using feats. Unlike 4e’s multiclass feats, however, which involved taking just one feat, and possibly taking the other class’s powers instead of your own class’s later on, Pathfinder VMC was a commitment to give up literally half of your feats for a series of locked-in benefits from the chosen class. In literally every case, this was a horrifically terrible trade, and no one should ever use a variant multiclass for any reason. Most of the feature sets available through VMC were barely worth one feat, let alone literally half of those you would ever have (barring bonus feats).
Reverted to a system very similar to 3e or Pathfinder, although because 5e is simpler overall various complications like skill points didn’t even begin to apply. Notably, multiclassing was officially an optional, “variant” rule, unlike previous editions. Also notable, each class had minimum ability scores you needed in order to multiclass into it.
Finally, each class provided some benefits—mainly proficiencies—that you only got if it was your first class, rather than one you multiclassed into later. This is similar to the maximized HD at 1st in 3e, which 5e also has, but potentially a much larger concern.
Again like 3e or Pathfinder, 5e supported “single-classed multiclassing” through things called “archetypes.” Rather than being external variants of a class, though, making surgical changes to its features (as in 3e or Pathfinder), the 5e archetypes were just class feature options that everyone who took the class had to choose at the appropriate level.
Many of these have nothing to do with other classes, just being different approaches to the class—for instance, barbarians could be frothing berserkers or contemplative shamans, wizards could focus on any one of the eight schools of magic, and so on. But several of them do make a character more like another class—barbarians could also become zealots, gaining divine power that might be reminiscent of a paladin, fighters and rogues could become eldritch knights or arcane tricksters, respectively, gaining some spellcasting, and so on.