"To be just, our law must be cruel."
- – Motto of the Adeptus Arbites
"One who trades freedom for security deserves neither."
- – Benjamin Franklin
"There are not enough jails, not enough police, not enough courts to enforce a law not supported by the people."
- – Hubert H. Humphrey
"There is no crueler tyranny than that which is perpetuated under the shield of law and in the name of justice."
- – Charles de Montesquieu
"I AM THE LAW!"
- – Judge Dredd
They are basically the FBI of the Imperium. But they aren't just any ol' cops, they are badass motherfuckers, think of a combo of RoboCop and Judge Dredd. They usually drag the criminal up to the judge who then ceremonially declares them guilty and being guilty means that you have committed a crime. And in the Imperium there is only one crime and it is HERESY!!!!!!! On some worlds though, they send criminals to Imperial Guard penal legions instead of executing them on the spot, thus they may receive the Emperor's benediction on the glorious field of battle. The Adeptus Arbites have slightly smaller balls of steel than guardsmen, they are on the other hand made of much more man sauce than SPESS MEHREENS and their sissy power armor.
The Adeptus Arbites have shown that social sciences, like the applied, have also been degraded in the 41st millennium while they help power Terra by using the remains of long dead criminologists to drive generators. Excessive brutality is not an effective way of combating crime as it builds resentment between law enforcement and the people they are there to protect from lawlessness. On the other hand, if the alternative is letting over-populated hives become horror fests of the worst sorts of crime, then it might be more cost-effective for the Arbites crack as many skulls as they like. Mind, they are still rife with crime, but it's mostly low-key stuff like drugs and
forbidden objects now those are really dangerous: xenos artefacts and the like are forbidden for a reason. Life is (literally to the Arbites) the Emperor's currency. Are you going to go on a serial killing and thereby waste the Emperor's currency? Not if you value your life.
If you want to include the Adeptus Arbites in a story or game of Dark Heresy, I suggest you read/watch/play/whatever the following for inspiration:
- Judge Dredd
- RoboCop (The original. The '80s and '90s knew how to make a dystopia look dystopian, unlike today's photogenic, scrubbed clean dystopias.)
- Die Hard
- L.A. Noire and/or L.A. Confidential (For that "law enforcement being full of corrupt douchebags" vibe)
It's important to note that NOT all Imperial lawmen are Adeptus Arbites (this point is stressed repeatedly in the Ciaphas Cain novels - many planets call their civil police force the Arbites out of convention). The Arbites are like the FBI and Interpol of the 40K universe, they take on the cases with interplanetary criminals or crimes against the Imperial laws instead just the local planetary laws. Some planets have only a single Arbites agent loosely overseeing all the planetary police. So feel free to write a story or play a game involving plain old police in spess and have the Arbites as an overseer/boss/meddling (and possibly corrupt) dickhead.
TL:DR: Arbitres don't give a fuck about regular crime unless it threatens the wider Imperium, i.e. can lead to rebellions, incursions and Imperial Tithe being unpaid.
What is the Law?
I AM THE LAW
The Arbites uphold Imperial Law, which is pretty broad. (It has to be, due to the varied nature of the Imperium and its myriad of different people at large.) It can however be very broadly summarized into two main points on a planetary level:
- Worship and remain loyal to the God-Emperor of Mankind
- Deliver the Imperial Tithe on a timely manner
Now, you might think: "that's it?", but the implementation of said points are often one hell of a pain in the behind to deal with and require frequent skull-busting. The first one consists mostly smoking out and cracking down on Chaos cults, Genestealer infestation, independentists, xenoists and plain old power-hungry morons wanting to take over a world, system, or sector through violent and tithe disrupting upheaval; or any other joyous thing that could fuck up a planet from the inside. They don't explicitly handle heresy though; that's what the Sisters of Battle and Ordo Hereticus are for (although they do help when asked). The second point of Imperial Law generally revolves around making sure that everyone and everything that is supposed to get on a ship gets onto ships: soldiers and/or materiel supplied for tithes, psykers in Black Ships, etc. They also keep an eye on Rogue Traders/Merchantsmen when they come in port (you never know what one of those shady bastards has and/or wants in their hold) and hunt down smugglers.
On a more galactic scale, the Arbites ensure that new worlds get used to the routine and older, important worlds give what they should and don't get complacent. They tend to be a little harsh because the step after the law is broken is full-on rebellion. The Arbites also serve as proxies for the High Lords, adjudicating conflicts between the Administratum and the Planetary Governors and ensuring their loyalty.
- Arbitrator Kazymanderas
- Good 6th Edition fan codex for the Arbites 
- 8th Edition fan codex for Adeptus Arbites, still in beta