In the early middle ages, law worked very differently than in modern society. We live all of our lives, today, under the law - if someone is murdered, we don't have a choice about what will happen: the police will step in and begin to investigate and bring the full weight of the law to bear on the case. In the early middle ages, however, the law didn't come to you - you had to go to the local authority and bring up your grievance and ask for a legal solution. It's much more like a modern civil suit - you had to take a criminal to court to get the law involved.

And the law was not the only normal, accepted mechanism for resolving disputes and seeking justice. Blood feuds and revenge killings were an allowed alternative for most of the early middle ages; some kings tried to regulate feuds (Alfred of Wessex, England, passed a law that tried to require disputants to attempt to settle disputes in court before they were allowed to raid each other's houses), but feuds remained an alternative and acceptable solution for resolving disputes through most of this period (and feuds, like laws, had certain norms and procedures disputants were expected to follow - it was an alternative justice system, not a breakdown of the justice system or lack of justice).

You periodically see kings try to force everyone to resolve disputes in the royally sanctioned law courts (for example, the seventh century Visigothic kings in Spain try to make it illegal to settle cases privately), but it took a long time for kings to get powerful enough to eliminate alternative justice systems. Some early medieval kings actually required subjects to try to sort our their problems locally and stay out of the justice system, presumably because they had too many cases and too few judges (this is one of the 10th c. english kings - Aethelstan, perhaps?). Eventually, most of these private forms of dispute resolution (like feuding) were suppressed by stronger kings exercising more effective and centralized authority, but some non-royal justice systems like Church law remained influential alternatives to royal justice for a long time.

A good introduction to this in England is Paul Hyams, Rancor & Reconciliation in Medieval England.

This is a fascinating topic in medieval Iceland, a state which lacked any form of executive authority yet was defined by its law, which was strongly enforced by communal action. Let me give you an example from a Saga, which are generally accepted to be reasonably accurate in displaying the law (as law was central to the stories contained therein and was a topic of common interest.)

So, Gunnar Hamundarson was a famous warrior and virtuous man, who gets embroiled in a feud. He is supported by Njal, a well respected man with an apt knowledge of the law, but for all intents and purposes he is one of the lower classes in Icelandic society, a farmer. Gunnar gets outlawed from Iceland at the althing, a sort of court where legal battles get decided and mediated upon by 'chieftans', powerful men supported by farmers. Lawspeakers, men with a knowledge of the law would say if any action were lawful or no.

So gunnar, after killing a few men, gets outlawed; essentially this means that he no longer has legal rights in Iceland, and so must leave. He is not being explicitly made to leave, but because he has no rights he can be killed and no compensation is required. It was a self mediating system; x amount of wealth, normally in wool, was paid for a man murdered; so without a police system killings were limited because they were expensive. If you did not pay, you were outlawed. Gunnar decides to stay and is promptly killed.

Generally they were investigated by the parties involved, with witnesses and juries. Punishment would be meted out by those aggrieved, within the confines of the law.

a famous quote from Njal sums up the situation: "for with law shall our land be built up and settled, and with lawlessness wasted and spoiled"

While this is by no means intended to prevent others from adding their input and insights, /u/idjet wrote a fantastic post on a similar topic about a month ago that could be of interest to you.

During Medieval Islamic period there was Police known as Shurta

"The police (shurta) as an urban force responsible for suppression of crime evolved from a military formation referred to as shurtat al-khamis, which apparently existed in Kufa during the rule of ‘Ali (656–660). The association between the shurta and the military continued during the Umayyad and early ‘Abbasid period."

According to an administrative text from Baghdad of the early Buyid period (c. 950) the chief of police had to be familiar with the scale of punishments, but legal education was not a qualification required from him. He was advised to study texts such as the Book of Brigands, apparently by Jahiz (776–869), and other unspecified works such as kutub ashab al-shurut wa-siyar al-muluk.

Police had to be familiar with punishments as early as 10th century Iran-Iraq region, they didn't have to receive legal education, there were even books subscribe to them before they enter the force. Another thing or weakness is when people interfere PLUS when their punishments were made public, a famous serial killer from Baghdad was brought in front of the Caliph to be killed, so sometimes rulers did interfere and it was kinda annoying but also a reminder and deterrent for the public.

Source: Medieval Islamic Civilization: An Encyclopedia p.622-623