This perk allows one to use any kind of power armor in the game. Prior to this perk, equipping power armor was impossible. This perk can be gained by two ways:
After gaining permission from Elder Lyons, speak to Paladin Gunny in the A-Ring of the Citadel. There is no charge or other requirement for training - the screen simply fades to black as Gunny begins teaching you. Once that's done, you may use any kind of power armor, including the Enclave and the Outcast variants. If you do not ask for training on your own, you will get it automatically at the end of The American Dream when you become an honorary member of Lyons' Pride, however these options appear very late in the game.
As in Fallout 3, this perk allows one to utilize power armor. Without this perk, any power armor found cannot be equipped, with the exception of NCR salvaged power armor. There are two ways of obtaining this training.
If McNamara remains Elder, wait 24 hours outside of the bunker for the lockdown to be lifted, then talk to him. Tell him you wish to join the Brotherhood of Steel and this will begin the Eyesight to the Blind quest. When this is completed you will then receive Power Armor Training.
NCR salvaged power armor does not require Power Armor Training in order to be worn, due to the absence of the servo-motors.
If you first complete For Auld Lang Syne and later complete Still in the Dark, Elder McNamara will be astonished that you already had Power Armor Training and will ask where you learned it from. However he then brushes it off as a result of the Courier being "well traveled" and will not give you the option to alert him of the former Enclave bunker nearby.
The Lone Wanderer and Courier 6 are the only people in both games who need a special perk to wear power armor, companions like Raul who would likely have no experience with power armor has the ability to wear it. Even raiders can wear power armor.
In Honest Hearts, there exists a dialog option enabled by this perk when talking to Ricky, where you can refute his suspicious claim that he shot a Brotherhood of Steel paladin through the eyeslit with an "11mm sub machine gun", by stating that the eye slits of the T-series are bulletproof.
If playing with the JSawyer mod, power armor training is no longer required to use power armor. Instead, wearing power armor with training gives a bonus to carry weight, equivalent to the weight of the armor.
↑The Courier: "You're saying that you killed a Brotherhood of Steel Paladin?" Ricky: "What's it sound like I'm saying? If I was saying what you said I was saying, then yeah, I said it!" The Courier: "Please go on. This promises to be entertaining." Ricky: "I was walking along, minding my own, and up pops one of them Brotherhoods. He yells "Hand over that laser rifle, asshole!" So I hand it over, just to make him think I'm scared! But really I'm not! I never am! Before he knows what hit him, I draw my 11mm machinegun and BAM! BAM! Right through the eyeslit in his helmet! D.O.A.!" The Courier: "What a load of crap. The eyeslits of T-series power armor are bullet-proof!" Ricky: "Then I guess this dumb fucker's armor musta been... D-series or something! All I know is, he died up real dead when I killed him, okay?" (Ricky's dialogue)