Initiative (role-playing games)
In most role-playing games and table-top wargames an initiative system determines in which order player characters and non-player characters take their actions, to avoid confusion on when a character gets to act. Methods of doing this vary from game to game.
Taking turns
In some games, such as Tunnels & Trolls, each faction takes its turn to act - and it is entirely up to the side acting who acts in which order for that side. This is one of the two most common systems in table-top wargames - with Malifaux having a variant of this by which each gang picks one member to act in turn, and the larger gang all acts together at the end.
Statistic based
The other common method, used by games like Dungeons & Dragons and Fate is for each character to be assigned an initiative number based on a relevant attribute plus a random factor, and for people to act in that order. Some games determine initiative order once, some once per round.
Shot Clock
Feng Shui and Arcanis both use a shot-clock; Each round is divided into segments or shots, and each action takes a certain number of these segments. Actions happen whenever the character's next free segment turns up.
Choice
In Marvel Heroic Roleplaying each character to act may choose anyone who hasn't acted already in that round as the next character to act. The final person to act in the round chooses who acts first in the next round.