Each player has territories marked with her color. In the mark there is a number indicating the number of armies in that territory.
When it is your turn the first step is to distribute new armies. In the status bar at the bottom you can see the additional armies you can deply. Each click on the territory you own will add one army to that territory (right-click to remove an army, mouse wheel works also). The number of armies you can add indicating each turn is determined by the number of territories you own and the continents you control when your turn arrives. In Addition you can get further armies by exchanging cards for armies.
During this time you can click on the "undo all" button to start the distribution again.
Once you distributed all the armies you have to click on the "ready for attack" arrow (or press enter).
Then select one of your territories to attack from and select one of the neigbouring enemy territories to attack to. Throwing the die and calculating the results will be done automatically.
You can repeat this as many times as you want or as long as you have more than 1 armies in a territory.
When you kill all the armies in the neighbouring territory you will move some of your armies to the captured territory. A pop-up window will let you pick the number of armies to move.