The Police are one of Atlantic City's biggest gangs. Unlike most other gangs, they do their business openly in public and have extensive ties to City Officials, Politicians and even The Feds. Their main role during the prohibition years was to locate and impound beer and liquor, which many officers then sell back to local crime gangs for a profit.
Your criminal actions will attract heat from them, and if they see you as a threat they'll eventually launch an investigation which can cause you to lose the game. You can use Lawyers to help keep them off your tail or hire Hitmen who are skilful at covering their tracks.
Should your gang fail a combat mission the police will usually jail any henchmen who fail to escape the crime scene. You can sometimes call a favour with a local Politician to get your gang member out of jail, but often you'll have no choice but to fight your way in to the prison to do a daring rescue.
If you know the location of Deputy Residences, you can bribe officers to improve relations with them (which may reduce rate of heat increase on some maps). As your relations improve you can buy enhanced weapons and even call a favour to reset the heat to 0 stars.
One of the most corrupt officers in the police force, Deputy Kingsley, is your main inside contact for the police. While you won't see him in combat missions, he will regularly offer resources and can help you lower the heat - for a fee, of course.