Ah, emergence. By far one of my favorite topics in games. Emergence, put simply, is the process by which patterns or developments emerge from a set of agents or entities interacting with a set of rules or laws. In nature, emergence occurs regularly. When entities interact with the laws of physics, strange and unexpected things can happen.
In the real world, the most beautiful, simple example of emergent behavior I can think of is swarming. There is no captain animal that directs a swarm. No, instead each individual agent follows a set of rules and swarming behavior often follows. Rules as simple as "stay within one foot of your neighbors".
In games, emergence is the process by which agents (players or otherwise) interact with a set of rules that produces outcomes not necessarily defined in the ruleset or explicitly defined by the game. This parallels nature quite perfectly, albeit games are much less complex. But like nature, emergence can arise from a very limited set of rules given enough agents acting upon a system.
The clearest gameplay example is one of physics systems. Again, this mirrors nature's emergence. In games, real-world newtonian physics are frequently simulated. Couple these "laws of physics" with the "rules of the game", and unexpected behavior can and does emerge. Deus Ex is a fantastic example here, where players can frequently use the physics system to their advantage in conquering the game. Moving and stacking objects to reach area not typically accessible or destroying sections of a level to advance, using explosives on walls to scale to areas at that point inaccessible areas are all examples of utilizing the game's simulation system to open up new possibilities. In addition, Deus Ex offers player choice in tackling challenges, offering a variety of lethal and non-lethal approaches to sistuations. The amount of choices the player has coupled with interacting with the well-defined system of rules in Deus Ex leads to emergent gameplay possibilities where players can discover and employ creative solutions for solving the game's problems.
Emergent Gameplay is so great because it's so much fun. A player discovering new possibilities within the rules and mechanics set forth in the game is one of the most rewarding feelings a player can have. Metal Gear Solid V, released last year, was heavily praised for the creative solutions players could employ to succeed in the game world, much in thanks to the care taken to craft the rules and mechanics in such a way that new gameplay could emerge natrually. Players were allowed to experiment with the mechanics, leading them to discover new strategies and mechanics not explicitly defined by the game world.
Emergence is not always intentional by the developers, but yields from systems which rules are developed in an open-ended way to foster interaction and player choice. Many people think emergence is the future of games and a concept we must embrace, and I happen to be one of them.