Warning, this is a heavy and thoughtful post. Do not read before having your morning coffee.
Emergence is the changing group dynamics and player interaction that happens during and in between games. This isn't a complete definition, nor is it even 100% accurate, but it's what I mean when I use the term. Episode 45 of Ludology spent far more time talking about this, and doing a better job of probing the depths of it than I could ever do. Ever since I listened to the podcast, I've been noticing emergence all over the place.
I think an example of this helps more than anything. The other night, I was playing Resistance Avalon with people who had never played before, or who had only played once. In our first game, there was very little table talk until I started pushing it. See, the first time I played Resistance, I didn't know enough about the game or the other people to feel comfortable with the direct accusations of "you're a spy" and so on, but this time we were all more comfortable with each other. In our second game, the table talk flew around for most people, and that was great. In the third game, even our quietest player was jumping in on the accusations and everyone was really trying to figure everything out.
Now, here's where you say something like "Well that's because Resistance is supposed to work that way." or "Players naturally do that when they are more comfortable with a game." Those are both good points, but that's part of emergent behavior.
Players will start to think of things on a higher level the more they play a game. Let me give you another example of a game. In Hanabi, players are working together to play cards in order without messing up. The catch is that you can see everyone's hand of cards except your own. On your turn you can only tell someone about a number in their hand, or a color in their hand. But what do you do if someone has say a 1,2,3 in red? Well, maybe you touch the cards in the proper order as you say "this, this and this are red." Some might say that that's cheating, but I call it emergent behavior. Both players have to understand the code. There are other aspects to that game that demonstrate emergent behavior, but let's leave that for the review.
In the aforementioned Ludology podcast, they use Go as a leading example of emergence. Since I don't know much about the game, I'll use Chess as a counter example. Chess has seen the rise and fall of strategies over the years. There is no perfect strategy, but there are strategies that can be hard to beat. The more you play against one opponent, the more you have to change what you do, else they would learn how to beat you every time.
Perhaps the simplest example is Rock Paper Scissors. Each part has something that it beats, and something it loses to. It leads to looking for emergence as you try to detect patterns in your opponent. What I'm getting at here is that emergence is all around us. It's one of the aspects that makes gaming great. Good games will lead to emergence. Strategies will rise and fall. Players will have to try something, fail, react, change, fail, alter, fail, try again, and perhaps fail. It can be a frustrating process, but cracking that code and seeing something for the first time can be one of the most satisfying experiences in life.
As it relates to game design, emergence is crucial. Sure, not every game has to have emergent behavior, but most tend to have some amount of it. Rules establish a framework for what we can and cannot do, but they often leave room for strategy and choices that even the designer didn't envision.
Take Tic-Tac-Toe for instance. The rules, start with a 3x3 grid, make a mark in a grid that no one else has marked in, connect 3 in a row to win. There's strategy there, and choices. Yes, it's been solved, but there is still the opportunity for emergence when someone learns the game.
The unexpected is often what makes life worth living. Trying to find something new, something that is in the realm of the constraints that have been established for us, but is something that no one has thought of before.