Favored Class is a games mechanic native to Dungeons & Dragons 3rd edition.
In a nutshell, Favored Class was developed from the principles of Racial Class Restrictions pioneered in Advanced Dungeons & Dragons: perhaps because of its origins, D&D has traditionally had an issue with pigeonholing races into classes based on what the designers thought were archetypical. For example, all gnomes will only use illusion magic if they study to become wizards, a dwarf would never use non-clerical magic, only a human has the purity of soul to be a paladin, and so forth. Needless to say, this idea was hugely contentious amongst the fanbase.
In recognition of that, when Wizards of the Coast took over the D&D license and decided to produce a new edition of the game, they decided to junk that rubbish - at least, partially. Fearful of arousing too much purist grognard rage, they created the system of the Favored Class, which would encourage players to stay closer to these archetypical classes. Working off of the new multiclassing rules, it basically functioned like this: apart from certain rare exceptions, all races had a favored class. If they had levels in this favored class, than either they needed to keep this class highest whilst multiclassing or suffer XP penalties, or levels in it simply did not count towards XP penalties, depending on whether you're talking 3.0 or 3.5.
The mechanic... wasn't exactly well-received. The whole idea of classes being favored bugged some fans, who complained that this didn't match their homebrew campaign's depiction of a given race, but most focused on the fact that it literally had no effect unless you were multiclassing and worked retardedly even when it did apply: You could take 1 or 2 levels in every class without penalty, but a character that took two levels in barbarian and four in ranger (both wilderness focused martial classes) did. Furthermore, it handicapped non-humans who were already weaker than the bonus feat equipped humans and required tracking everyone's experience separately, which was a huge pain in the ass for no real benefit. And thus, the mechanic was cancelled with 4th edition's release, and doesn't look like it's returning any time soon.
RPGA added an exploit to these rules. RPGA kept high level characters in limited numbers by retiring them once they hit high enough level. By extensively multiclassing as you got near the level cap, a character could acquire 100% EXP penalty and never risk hitting the cap. This allowed a character to play through infinite scenarios and build power by acquiring theoretically unlimited wealth instead of a higher level (though for caster preferred to abuse spells with an XP cost).
However, Pathfinder First Edition had its own variation of the concept. Instead of placing some arbitrary XP handicap on various classes for each race, PF Favored Classes instead allowed for each race to gain a particular minor bonus for each level they take in one class (two for Half-Elves because snowflake), which vary in usefulness from more spells and extra class features to...a situational bonus for a skill test. This is hampered further by the rather vast array of races available to Pathfinder, of which about 60% of the non-core races will lack Favored Class Bonuses for certain classes because Paizo can't be arsed to write for all of them. Of course, these guys aren't entirely SOL as that Class Bonus could still be used for a skill rank (which is pretty much a crutch for classes with abysmal skill ranks like Fighter and Cleric) or a spare point of HP. (a crutch for the classes with low hitdie) It wasn't as widely despised as WotC's iteration, but Paizo's lack of coverage and the flaws in the construction of other things for the classes did hamper the mechanic.
Come Second Edition, Paizo would similarly discontinue use of Favored Class, if only because there was no way for them to implement it as a feat.