Achievement Mode is a house rule based off the achievements on the iOS game. They can only be played on maps currently on iOS. All you need is a list of the achievements, which can be found on the app, and on the Achievement page
At the end of the game, if a player has completed the necessary things to complete the achievement (bar winning), they receive 20 points.
Some of them are impossible, like Railway Tycoon, or hard to measure, like Economy Class.