In a mass-market game, that wasn't really much of a question, because in Monopoly, Scrabble, Chess, or any number of other traditionals, the answer was always obvious: the next player.
However, as games continue to evolve, adding on new levels of complexity, the answer is becoming more difficult, and the question of who goes next isn't always so obvious.
Starting with Card GamesI would suspect that card games were really the first ones to raise the "Who Goes Next?" question. This is because most card games divide up their play into tricks, where each player plays one card. This means that if you kept having the "next player" go next, then the same player would lead each hand, which would allow him to play to his strengths and win most of the time. Because of this large first-player advantage, card games developed two different answers for the "Who Goes Next?" question, based around their two core units of card play.
At the end of each trick (or each round of play, if you prefer), the player that wins the hand is usually the one that leads the next hand. This tends to offer two rewards to winning a trick: getting the cards of the trick and leading off the next one. Different games play this off in different ways. In most card games, from Bridge to Spades, taking the trick is crucially important, while winning the lead just lets you control what happens next, but in a game like Gang of Four there's no trick taking, and so all that's important is getting to lead the next time around--underlining the importance of this whole exercise.
At the end of each hand, the lead tends to shift around the table clockwise.
And that's how almost every card game answers the question of "Who Goes Next?". As we'll see, some board games have adopted these same answers, but more recently we're seeing a lot of variation as well, going beyond card games' staid solutions.
Back to Board GamesAs I've already said, in Ye Olden Board Games, you just kept playing sequentially. There's still a lot of designers that use this model most of the time, including stand-outs like Reiner Knizia.
However, we're increasingly seeing games designed around limited choices and limited resources. These games also tend to divide up into rounds (or tricks, if you prefer), with the most common round giving each player one opportunity to play--though they're sometimes much longer. At the end of each round, a new player then gets the opportunity to be first.
Puerto Rico is one of the earliest notables in this area, but the style of play has really exploded in the last several years with more role-selection games and more scarcity games, all of which tend to make it really important when you go.
Puerto Rico took the old card play model of shifting the lead one to the right each round, which is a fine, simple method. It can also be incredibly painful when you go from first to last, because that increases the length between your turns. It's not a big deal in Puerto Rico because of the low down-time, and in fact becomes a core element of strategy. However, it drove me crazy in Eagle's Attack! (with expansion) because the down-time was high enough that you'd have over an hour in-between your first and last turns.
I'm somewhat fond of the retrograde turn order, where the first player goes counter-clockwise rather than clockwise. Thus, in the extreme case, rather than going from first to last, you instead go from last to first. However, it's more confusing, and can cause serious balance problems when a player gets two turns in a row.
Complexity in OrderingIt's been several years now that the number of round-based board games have been increasing, and I'm pretty pleased to see that the thought being given to the whole question of "Who Goes Next?" is increasing too.
After moving the first-player clockwise (or counterclockwise) I think the next easiest answer is to allow the player with the lowest score to go first (like Torres does), if being first is an advantage, or to make the player with the highest score go first, if being first is a disadvantage. Thus, you have a simple rule that doesn't continually advantage the same player and you have a catch-up mechanism, all in one simple rule.
However I like the board games that have integrated the whole question of "Who Goes Next?" into the game play even better. They treat it as yet another resource that you have to manage.
Caylus and Pillars of the Earth both made the first-player ability a resource that you can purchase, but at the cost of doing something else useful, which is an OK, but kind of in-your-face answer.
I prefer the solutions suggested by Phoenicia and Cuba. In the first you go first in the next round if you have the most victory points, but by doing so you're probably limiting your resource production. In the second you go first in the next round if you choose your roles in a specific order. In both cases you have to minorly disadvantage yourself in other ways to increase your probability of going next, which is generally a more nuanced and interesting answer than just choosing whether to take the first-player action or not.
ConclusionI've thus far left out my favorite "Who Goes Next?" mechanism, which is that found in Thebes. Therein, every action you take costs time, and whoever has spent the least time overall goes next. This is neat in several ways, because it turns the "Who Goes Next?" question on its head, because there's no longer a guarantee of getting equal turns. Instead, turn management becomes pure resource management.
I suppose you could say it's another way of looking at an action-point system, in that you can take one big action or many small actions, but if so it's more clever and less likely to cause analysis paralysis than any action point system I've ever seen.
Thebes also suggests that we've just scratched the service of the "Who Goes Next?" question. As designer games continue to evolve, we may see lots of other solutions to this question that was taken totally for granted back when we just played Monopoly and Battleship.