Raider Attacks are a specific random event that happens occasionally during regular play. It's very obvious, as an alarm will begin blaring (as with any negative random event) and you will see the scum running down the path to break down your door.
During this siege, the raiders will begin depleting your resources. These attacks will get progressively more violent and difficult to defend against as you get more dwellers and defeat more raiding parties, but luckily I seemed to easily defeat them as I got around 15 armed dwellers to fight back.
The main defense is arming your dwellers. You can do this by selecting them and giving them weapons and outfits to help combat the threat. You should work on equipping everyone in your vault at all times, this will help against any kind of disaster that may occur and will keep your storage from filling too quickly.
The Raiders will move from room to room, attacking anyone in their path (excluding pregnant women and children, the saintly knights they are). From my experience so far, they will not stay in a room if there is no able-bodied resistance in the room with them. They will stay to fight whoever happens to be in the rooms they enter, but will only stay for a short amount of time until they decide to move on, regardless if they killed anyone or not.
The raiders move from floor to floor, before reaching the bottom. They will fight in the last room that has warriors and then they will flee. If you do not kill them before this happens, the Raid will not count as stopped.
You can put well armed dwellers in the first room of the vault, called the Vault Door room, to assign them as guards for Raider defense. In my case, I do not worry about it too much. I have (at the time of writing) 30 dwellers and all of them are armed decently well, so the raiders never make it more than 5 rooms into their siege.
If you want to check your progress, there is a counter under the stats page for Raider Attacks Stopped. It's located under the "Cumulative Since (Current Date)" header in the Stats page.
EDIT: I have been playing on the Windows 10 version of the game, and have been idle'ing for some of my Raider Attacks, which seems like a decent way to farm them. Note, that if you idle you run the risk of multiple random events damaging characters, with the result of some of them dying in raids or fires, ect. Keep an eye on the vault when you hear the siren blaring.
Also, it would seem that the more vault dwellers you have, the higher likelihood that you get a different assault, albeit from Ghouls or Deathclaws. This will unfortunately just take more time away from you, and I don't know how to exploit Raider only attacks as of yet
People are recording Feral Ghoul attacks as low as 40 Dweller population, so you might want to try grinding this achievement out with >40 population before going for the higher room achievements.