For most of my life, a "game master" was the person who ran a game of a roleplaying game. Specifically, it was the "generic" and legally-distinct version of the D&D term "dungeon master." Once I really started playing tabletop games however, I discovered that many games had a kind of game master role built in, and with lots of different names. The concept was always the same, though. A game master was the person playing the game world, in contrast to the people playing characters. The longer I've been playing tabletop games, however, the more I've come to realise that the concept of a game master is broader than even that. In fact, my personal definition of game master has expanded a lot lately, and sometimes even in a game that has no game master, a game master emerges. Here are some of the different kinds of game masters I've identified so far.
The first kind of game master is the traditional and, as far as I know, the one the term was originally intended to describe. An ordinary everyday game master is the person who runs the dungeon in contrast to other players who control just one character, or maybe one character and a handful of henchmen or a familiar or some kind of companion.
A game may have special terminology for this player, such as "Keeper" in Call of Cthulhu or "Zargon" or "Morcar" in HeroQuest. Whatever the terminology, the functional role is consistent: A game master starts the game with free access to the map, history, story, and tactical statistics of the game world, and manages all the events that happen around the player characters.
Combat in roleplaying games is often fast enough to be only a subset of the time spent in the game, but in wargames combat is the game. It's not uncommon for wargamers to connect several combats with a continuous story, although unlike a roleplaying game the story isn't generated by players interacting with non-player characters. Instead, the results of each combat influences the state of the world, and each army involved.
The game master for a wargame manages the story of the campaign. The game master might invent a dynamic storyline that evolves based on combat outcomes, or the game master may simply know an unchanging story (fictional or non-fictional, depending on the genre of wargame) that influences the condition of a battlefield.
In some cases, the game master for a wargame may serve a similar role as the RPG game master by actively controlling an additional army, or non-combatants caught on the batlefield, or environmental effects, and so on.
It wasn't the original intent of the role, but the game master in tabletop roleplaying games is often also the person who knows all the rules. Then and now, there is usually a rulebook specific to general players and a rulebook for the game master player, so it's very clear in nearly every RPG system that all players are meant to learn the rules. In practise, I think many a game master is also the primary recruiter for the game, so it's common that the game master already knows the rules and is willing to introduce other players to them. The games master is also the rules master.
This tendency also serves the game. It's useful, in a game that requires a game master, for there to be a single source of truth. When a question about a rule comes up during the game, it's a matter of expediency that the game master has the final say. A ruling is made, the game continues.
Interestingly, even in a game that doesn't require a game master, it's not uncommon for one player to know the rules better than other players. This person is sometimes the person who owns the physical assets for the game, or it's the person who's had previous experience playing the game. When that happens, this person acts in many ways as a game master. This person might teach other players how to play, and also gets to make decisions about how rules apply to a specific situation. It's not literally a game master role, but functionally that person is the game master.
The role of a game master isn't always an official position. In a way, if you're a person who's just happy to try a new game, and then inspires others to join in, then in a way you're embodying at least an aspect of a game master. You might not actually be playing the part of a game master, and in fact the game in question might not even have a place for a game master.
When you're driving a game by virtue of being comfortable with trying something new, you are serving as the "driver" of the game. That's not everything that a game master traditionally does, but it's where a master of ceremonies and a game master intersect. You're the catalyst for game play.
The catalyst for game play isn't always the same person providing the foundation for the game. Sometimes an aspect of game mastery is embodied by the person doing the physical and mental work of just keeping game assets sorted and available. This person is a kind of librarian or secretary of gaming, and is often found assisting players of wargames or big complex games with so many components that it reasonably takes a person dedicated to the task.
This is the person apps try to fill in for, and the absence of a librarian is the reason that some tabletop games are easier to play as computer games instead. There's a genre of modern tabletop games that secretly expect you to have a whole person just to maintain stacks of cards and envelopes and trays of tokens and paperwork and rules references. Without that person, the game play is possible but painful. It might not seem like the game librarian is running the game, but the librarian is literally powering the engine that makes the game go.
The term "game master" is something gamers invented to describe a specific role, but I think as games have evolved the opportunity to reflect the role has broadened.
I don't consider myself "game master" when I'm running Mansions of Madness for friends. All I'm doing is reading stuff off the app, clicking some buttons, dealing some cards. It looks an awful lot like being a game master, and in the end I think it actually is being a game master, except the storytelling part of that role happens to be unnecessary. It doesn't make you less of a game master just because you don't do everything a game master can do during the course of a game.
I think the concept of a game master is almost universal in modern tabletop gaming. Somebody takes the lead, even if only by owning the game in the first place. It might lead someone to taking an active role as a game master in other types of games, or it might just be a nice way to acknowledge that someone has taken a little extra effort to make a gaming session happen.
I imagine in some fictional stuffy old social club, there's a Latin inscription over the door reading: To become a game master you must first master the game. I don't know what it means, but it feels important.