With Paralyzing Palm, you will sometimes perform a special V.A.T.S. palm strike that paralyzes your opponent for 30 seconds. Note that in order to perform the Paralyzing Palm, you must be using bare hands or an Unarmed class weapon, such as the deathclaw gauntlet, spiked knuckles, power fist, etc. The chance of paralysis is 30%.
In Fallout: New Vegas, this perk essentially allows a character to do a more-powerful version of the Ranger Takedown inside V.A.T.S., as Ranger Takedown can only be done outside of V.A.T.S.
The Paralyzing Palm is a chance event that may occur during a V.A.T.S. sequence. It will replace one of the player's Unarmed attacks in V.A.T.S. with a Paralyzing Palm strike. The V.A.T.S. sequence is not interrupted, and the player will continue to attack the paralyzed enemy if there are more attacks queued. It does not cost any additional AP, or do any more or less damage than it would have done if it were a normal strike. It has a unique look when the player is actually unarmed (the "striking" hand is an open palm with curled fingers while the other hand is held open) but appears as a normal strike with unarmed weapons.
The Status screen that tracks the number of Paralyzing Palms performed calls them 'Paralyzing Punches'.
pcxbox360ps3 Attacking a paralyzed enemy with an Unarmed weapon can be difficult at times. The hitbox that determines where the target can be attacked for damage can sometimes disappear altogether. Attacking such a bugged creature will result in specks of dust coming off it with a dull thud sound, as if the player were attacking a wall. [verified]
pcxbox360ps3 If an enemy is paralyzed, and then shot with a Gauss rifle using a second V.A.T.S. attack, there is a chance the enemy will fly off into the distance from the knock back of the Gauss rifle. [verified]
ps3 It seems to be impossible to paralyze ED-E using paralyzing punches, as they'll never trigger when attacking him.[verified]