We're the Regulators. We've dedicated our lives to bringing the evil to justice. And out in the Wasteland, there's only one brand of justice: the gun.
”— Sonora Cruz about the Regulators
The Regulators are a group of vigilantes who wear cowboy-style hats and dusters in the fashion of pre-war gunslingers and outlaws.
Regulators wear Regulator dusters and use laser weapons (laser pistols and laser rifles), although more commonly they are equipped with combat shotguns, Chinese assault rifles and combat knives. In terms of damage output, they are about equal to a Talon Company merc, although they are easier to kill because their dusters have between 2-5 DR while Talon Company mercs wear 16-26 DR armor. Regulator patrols typically operate in three man squads when searching the Wasteland for the Lone Wanderer, if they have at least evil Karma. They are no more deadly or effective than their evil counterparts, the Talon Company mercs.
The Regulator's main base of operation is Regulator HQ. It is only accessible with the Lawbringer perk in posession.
Unlike their bad Karma counterparts, Regulators target the Lone Wanderer if their Karma ever goes to "Evil". It is also likely for a "Good" player character to encounter hostile Regulators if their Karma has previously dipped into evil territory. This can result in squads of Regulators appearing alongside squads of Talon Company mercs seeking to fight them.
Regulators carry bounty notices against the Lone Wanderer, regardless of Karma.
If you have evil Karma, the Regulators will come for you. They will usually spawn nearby your location. Below is a list of where they may appear if you fast-travel to that location. Regulators will not respawn after they have spawned in one of these locations, their bodies will also lie there indefinitely.
Regulator is the actual term for an official who maintains control and supervision of an area of public interest.  As a result, many armed groups, both vigilante and official, have been named "Regulators" throughout American history. One of the most notable groups in American history were the Regulators of the Carolinas around the time of the American Revolution.
The following is based on Fallout 3 cut content and has not been confirmed by canon sources.
pcps3xbox360 If the PC has access to the Regulator HQ and murders all of its Regulator inhabitants, Sonora Cruz will still remain friendly towards you as if nothing had happened. [verified]
ps3xbox360 Sometimes the Regulators will go through their dialogue as if they are hostile, but upon completion, they do not fire and appear friendly. They do not move or in any way pursue the player. If fired upon they do turn hostile and open fire. [verified]
ps3xbox360 The regulators sent out after the player can have skin problems where the face is of a Caucasian and the arms are of an African American. [verified]