The Fat Man is a missile launcher that fires mini nukes, and you can grab it just after leaving Vault 111!
When you first leave the vault, head directly east for about 2 minutes. After navigating around a small lake, you'll see a junkyard symbol appear on your compass. Head towards the direction of this icon, and you'll soon discover the Robotics Disposal Ground. Hidden among the heaps of scrap, you'll find a fusion core and ye olde Fat Man. No ammo unfortunately, but you're one step closer to causing some serious carnage.