Partially quoting myself from previous answers to similar questions:

Most societies have had people who's duty was to "keep the peace" so to speak.

In Ancient Greece, for example, city states had publicly owned slaves who acted as a police force. They kept order and controlled prisoners, but they didn't investigate crimes. Most crimes were considered a private matter so the citizens dealt with them themselves.

The city of Rome never had a real police force, though it did have a force of men set up for the protection of each ward. These men were responsible more for public order and physical protection and had no role in criminal justice.

Medieval European cities had forces along these same lines. For example, the duties of Lieutenant General of Police, the foundation of the Paris police, were described as "ensuring the peace and quiet of the public and of private individuals, purging the city of what may cause disturbances, procuring abundance, and having each and everyone live according to their station and their duties".

The thing to keep in mind about criminal justice as a whole is that, for most of history, crimes were considered private matters between individuals. With crimes being private matters, there was no impetus for the state to provide police as we know it today - they only needed a force that could keep order. Eventually a new idea took hold in England: violating the King's Peace (through murder, theft, etc.) was not just a crime against an individual, but a crime against the whole kingdom. This brought about the prosecution of criminals even when no private citizen brought a case against them.

This eventually led to organizations like London's Bow Street Runners, founded in 1749, who are regarded as the foundation of modern police. They were a formalization of the existing practice of 'thief-takers' - people who solved crimes for a fee. Then, in 1829, in response to the growth of London (and the growing difficulty in keeping the peace) the Metropolitan Police Service was established. This was the first professional, modern police force that was built as a civilian service, with civilian oversight, and accountability to the people.

So the idea of "city guards" is, on the face of it, fairly accurate. There weren't police, as we know them, only a force of men set up to prevent riots, keep unwholesome people out of the city, and maintain order.

I can tell you how they did it in England - a combination of devolved administration and sheriffs.

The sheriff was a crown official responsible for enforcing the king's laws in a county or shire. This position went back to the Anglo-Saxons, who appointed a 'reeve' in each shire for the same purpose. Over time the title 'shire reeve' was contracted to 'sheriff'. The Normans kept this institution because it was efficient.

The sheriff was both responsible for detaining criminals and putting them on trial, though for most offences they could not also act as the judge, and would have to request one from the crown in preparation for court sessions, which were usually held monthly. They had authority over the local aristocracy, though often worked with them to keep the peace. Many manor houses also had their own reeves. Sometimes this partnership with aristocracy could go horribly wrong - in 1189 in York, the sheriff brought in local nobility to help quell an anti-Semitic pogrom, but they joined in. Most of the time, however, the relationship between sheriffs and local aristocracy was amicable and productive.

However, a sheriff is only one man, and were often responsible for multiple counties at once. From their inception in the 9th century, they had the authority to raise military forces, the posse comitatus, for law enforcement purposes. Often this would be a small force for individual arrests, but at times it could be a moderately powerful army and in times of war they were expected to act as recruiters. Each county or shire was divided into Hundreds, and each hundred had a deputy to the sheriff called a bailiff, who handled minor crimes at a local level. Beneath the bailiffs were the constables, who handled crimes at an even more local level, down to individual settlements.

Working with the sheriffs, bailiffs, and constables were the coroners, who were the investigators. For complex crimes like murder and theft, they gathered evidence for the trial. Usually they reported to the bailiffs, but each shire had slightly different command structures depending on how much oversight individual sheriffs and bailiffs wanted over the process.

This is where a system of devolved justice came into play. Many towns had their own courts and systems of law enforcement. These could work in tandem with the crown sheriffs, but could also be almost entirely independent. Many cities and noblemen were happy to let the sheriffs conduct business in their land, even if they were not under obligation to do so, but others kept the sheriffs far away. The best documented independent system is London's. London was divided into wards, each run by an Alderman. The Alderman was responsible for keeping the peace in their ward (along with everything else) and had the authority to raise a militia for patrol duty and to make arrests, much like the posse comitatus, though the forces of these Aldermen were usually much smaller. Depending on the crime, suspected criminals would be referred to one of the many courts of the city. London did have a sheriff, but it was a largely ceremonial position and his main responsibility was not law enforcement but tax collection. FOr most of Medieval London, the sheriff was a Londoner anyway, and where they did get involved in law enforcement it was within the legal framework of the city, not the crown. When Richard II tried to give the sheriff of London more powers over Londoners in the later 14th century, Londoners pushed back and there was, ironically, mass unrest over the issue.

So let's say someone has stolen and slaughtered one of your sheep in some English farm. You would complain to the local constable. The constable would then go to the bailiff or lord's reeve depending on where they were. They would then send a coroner to investigate the crime and gather evidence. After finding out who did it, the bailiff or reeve would gather a local posse and arrest the culprit. They would then be detained until the lord or sheriff next held court, which would usually be no more than a month away.