"For every action, there is an equal and opposite reaction."
-Newton's Third Law
Sorry, Isaac. Not necessarily in D&D.
Like morale, reactions rolls have also faded from the later editions of the glorious mess that is Dungeons & Dragons. Sure, there are skills and such that allow for things like diplomacy and bluffing, but I am not a fan of too many defined skills in D&D. It's fine in games that were designed from the start for them -Runequest/BRP springs to mind- but they as much a hindrance as a benefit in old-school play (whatever that means).
Back on topic, the encounter reaction rules in versions like BX are incredibly simple. Other than the Charisma adjustments table, they are one paragraph of text and one 2d6 table of five possible results (B24). The rules simply state that while some monsters will nearly always behave in the same way (e.g. mindless undead attacking), it is possible for some creatures' actions to vary.
I've covered a bit about hirelings in an earlier post, so I'm mostly sticking to the encounter mechanics here. Suffice to say, the retainer reaction rolls (B21) help add a layer of detail to an integral part of the BX-style game. Adventuring parties were assumed to include meatshields and the like. Uncharismatic PCs, miserly offers of pay, or poor treatment could make life difficult for a party that needs to pad its ranks or flesh out weak spots in its lineup.
As stated in the preceding section on party actions, if they choose to talk, they might influence the monsters' (or NPCs') attitude toward the encounter and, by extension, the PCs. In my games, I interpret "talk" loosely. A common language isn't always required. An offer of food to a predator can be as effective an overture as speaking confidently and calmly to a goblin patrol in their tongue. (Side Note: This can be a case for using alignment languages and making those INT bonus language slots worth something in one's game!)
We didn't deserve you, Steve!
A couple of caveats to consider. One, the DM always has the right to veto making a reaction roll and simply decide how monsters, etc. will act. An elven patrol is unlikely to let the party go after they just burned down the Sacred Oak, no matter how charismatic the PC spokesperson is!
Two, even if a roll is made, there could be negative modifiers. The goblins mentioned earlier might be fine with avoiding a fight under normal circumstances. However, if the PCs are there to stop their shaman from performing a blood sacrifice that will give them victory over the villagers, the patrol just might be less inclined to believe the party is "just passing through." Even unintelligent creatures can have circumstantial biases. In real life, encountering a bear in the woods will usually not result in an attack (if you aren't stupid about it, that is), but a grizzly sow with cubs can be another matter! Tossing her some iron rations is probably not going to cut it, even for evasion purposes.
I guess what bothers me about this mechanic falling by the wayside is it removed a big incentive for actual roleplay in a dungeon environment, as opposed to just chatting up the tavern wench for -ahem- "rumors." To me, the BX reaction mechanic was an elegant solution that allowed for player agency and a bit of luck.
I confess I don't know much about 5th ed. What little I've played of it didn't seem to lend itself to this, but I could be wrong.