Your automatic weapons now do double damage and have a greater chance to stagger opponents.
Ranks 2 and 3 each add 10% hip-fire accuracy, for +20% total. Rank 4 gives a 4% chance to stagger a target. Rank 5 increases this to a 6% chance. In addition, if Nuka-World is installed, Commando now becomes a crafting perk, mainly for the automatic variant of the handmade rifle.
In general, Commando affects any automatic pistol or automatic rifle, but, despite its wording, does not affect automatic heavy weapons. There are also exceptions: for example, a pistol-gripped plasma thrower is automatic, but is affected by Gunslinger, not Commando.
Because it is a multiplicative bonus, it effectively negates the penalty of the Fast Shot trait in Fallout: New Vegas when using two-handed weapons (1.25 x 0.8 = 1).
The above lists are not comprehensive. In Fallout 3 and Fallout: New Vegas, the perk does not use a list of weapons to apply the bonus, it simply looks to see if the animation type used in V.A.T.S. is either that of a 2 handed rifle or 2 handed automatic weapon. If it is, it gets the perk bonus.
In Fallout 4, it seems any weapon with an automatic receiver receives benefits from the perk, so if you turn a pistol into an automatic pistol, it will get the benefits of the perk and lose the benefits from the Gunslinger perk.
The rifle in the Fallout 4 and Fallout: New Vegas perk images appears to be an R91 assault rifle from Fallout 3, which does not appear in those games.
In Fallout 76, even though the perk benefits fully-automatic guns, the perk cards artwork shows a bolt-action scoped rifle similar to a hunting rifle.