So, you want to know how the laws are enforced in the magical world! Want to know more about how it's built/functions. Well, here's the place! For an easy, simple answer. We try to keep the magical law enforcement as close to non magical as possible! So you will see many similarities between magical law, and muggle law.
AURORS
Aurors are one of the most widely recognized law enforcement group within the wizarding world. Aurors are, for lack of better words, the magical police force. They enforce magical laws, arrest criminals, stop crime, etc. Aurors are a universally used term by the magical community, so most magical governments will refer to their main police force as "aurors". Aurors keep the day-to-day peace in the magical world. There can be special divisions in an Auror force, just like a police station, where certan aurors can specialize in certain types of crimes. There are even aurors who specialize in combatting dark wizards. Overall, think of them like police.
In terms of numbers, this will be very similar to regular police forces. There is roughly 2 auror to every 1,000 witches/wizards.
Auror training takes 2 mandatory years, and can be quite difficult to pass. Auror training covers a wide variety of subjects, and skills. The most commonly used skills aurors have and are trained in are: Defense Against the Dark Arts, Transfiguration, Charms, and Potions. But to get into the Auror program you must have five N.E.W.T.s of no less than 'Exceeds Expectations'.
HIT WIZARDS
Hit Wizards function similarly to "special ops" or "SWAT". Basically, they are similar to Aurors in some ways but they do not deal with the day-to-day maintenance of order. Instead, Hit Wizards are highly specialized in high risk situations. You will see Hit wizards used in hostage situations, bombs, raids, and hunting down very dangerous criminals. Hit Wizards often times have more leniency in terms of what they are allowed to do "on the job" as their job often requires them to think quickly on their feet and they rarely have time to seek approval for things from higher ups. In other words, if Hit Wizards are called into a situation they are always given the order to stop the situation at any means necessary.
Hit Wizards are fewer in numbers then Aurors, with only 1 Hit Wizard to every 10 Aurors.
Similar to Auror training, Hit Wizards require 2 years of specialized training. They also have the same N.E.W.T. requirements as an Auror to get into the training program.
MAGICAL INTERNATIONAL CRIMINAL POLICE FORCE
Magical Interpol is a world run police organization. Unlike Aurors and Magical governments, Magical interpol's jurisdiction doesn't even if you cross a border (unless it's into a country not part of International Confederation of Wizards. Most countries are a part of it, though). Magical Interpol is not like aurors however, in the sense that they don't enforce day-to-day laws. In stead, Interpol enforces international laws. While they may be called in to assist a local government, it is not their primary job. Interpol often works cases in which criminals have crossed borders and either need to be extradited to a country, or when a criminal is "international" (as in operating and committing crimes in more then one country).
Magical Interpol is very small and elite. They're are only around 500-1000 employees around the globe.
Magical Interpol requires similar N.E.W.Ts as Aurors and Hitwizards, and training takes 2 years. In addition, however, Interpol agents are required to speak more then one language (since they operate in multiple countries), and often will learn more languages as they work.
WIZENGAMOT
The Wizengamot is the judge and jury of the magical world. They create, enforce, and change laws. They are also the ones crimes are brought before for judgement. The Magical world does not use a jury the way the US does, however, and is instead the Wizengamot, as a chamber, is your jury. For a conviction, the Wizengamot requires 2/3 of the Wizengamot's votes. They determine, if convicted, what the punishment for your crime is as well. They can give out fines, prison time, and even death sentences.
While the Wizengamot does not require any specialized training, it is most commonly made up of former aurors, hitwizards, lawyers, and obliviators as they are the most familiar with magical law (since it was their job to enforce it).
EXECUTIONERS
Ah, the executioners. While most crimes in the magical world do not require this individual, the magical world does still believe in the death penalty. So, the Executioner is an individual that is appointed by the Wizengamot to perform these death sentences. The Executioner isn't limited just to wizards, however. They can be called in to put down dangerous captured creatures and other magical beings in custody.
There is usually only 1-3 executioners appointed at a time.
No special training is required to be given this position, but executioners must have nerves of steel and undergo intense psychological testing, as their job is surrounded by so much death.
AZKABAN GUARDS
Thankfully, the death sentence is not a common punishment. More often, you will get a fine or prison sentence. If you are given a prison sentence, that's where these individuals come into play. The individuals are in charge of the prison and all prisoners. As Azkaban no longer utilizes Dementors, they play a vital role in day-to-day prison life.
While no special training is required to be a guard, Azkaban guards are often times former Aurors and Hitwizards. Either that wanted a less dangerous occupation, or due to other circumstances are no longer able to work in the field.