The Alien Invasion is a competitive event that will remain open every two weeks during a specific period of time. Check the countdown in the red event widget (left side of the playing screen): it will appear some hours before and it will be greyed out until the event starts. 
The main goal is to train as many recruits as you could during the Alien Invasion. You will get special rewards for your regular prisons, according to your position in the ranking at the end of each event. Every time an event starts, all the progress in the previous challenge will be also restarted, and you should start each challenge over to win the proper prizes.
On the right side of your gaming screen, you will find 5 golden badges that will indicate how many in-game days your recruits need to finish their training. Upgrade and max out all the items in your military camp to earn new badges, and reduce the training time.
Take in mind that the last golden badges will be unlocked thanks to the Seargent, the Weapons Workshop and the War Simulator, which should be paid by gems. Please also take in mind that Extra Barracks won't affect to the golden badges' progress since they also need to be unlocked with gems.
You could also earn Medals of Valour inside the Tactical Chests: you will unlock permanent upgrades with these event chests which will help not only in your events but in all your prisons. The chest will always provide a fair amount of card fragments. The Medal of Valour could appear in any of these chests as an extra card. These medals are cumulative and will provide 100 extra recruits to your counter automatically. You can accumulate as many medals as you wish, but they must be spent during the ongoing event, or they will be lost.
Managers purchased in your regular prisons WILL work during the Alien Invasion, as well as the no-ads bundle. Special items paid by gems during the event (SERGEANT, EXTRA BARRACKS, WEAPONS WORKSHOP AND SIMULATOR) will only be available during the event where they were acquired.