I am trying to create an online chess game using ActionScript and Flex. It works well when I have two players on the same PC, but I want to publish it as a web app. How can I transmit a move or action from one user to another, such that the other user sees what the first one did?
You will need some kind of server in the middle to manage the game state and transmit game data to the players.
The server's role is to take a player's move, make sure it is legal, update the board state based on the move, then push that change out to the other player so their game updates.
As far as technology goes, check out SmartFoxServer (http://www.smartfoxserver.com/) - something like it would do very well as your back-end technology.
Additionally, a quick Google search of "open source chess server" shows quite a few of them exist. You might be better off leveraging someone else's work. Remember, the server's job is not just to pass messages between game clients. It also has to confirm that all moves are legal, and that players aren't trying to cheat by sending illegal moves.