I will preface this with caveats: My knowledge is about England, and London specifically. London is a very large city; smaller cities and towns, so long as they were chartered, would likely have been governed by similar parameters, except on a smaller scale.
London has a long history--it was a city during the Roman age and was particularly important because of the bridge built there--the first one inland along the Thames. The city declined after the Romans withdrew and was reduced to a small settlement (the Thames being a boundary between hostile kingdoms) but by the 9th century or so the site had been reoccupied and eventually the bridge was rebuilt. This is the period that London acquired its traditional "freedoms" --that is, the ability to elect its officials, jurisdiction over taxes, weights and measures, and local security. After the Norman Conquest, the Crown gained some control over the city by granting London a charter with the same rights they had traditionally considered "freedoms." Royal assent was also required for the elected Mayor and Sheriff of London.
This is how it worked: The city was divided into Wards, each which had an assembly of all free men who met a certain level of income. Ward assemblies appointed various officers such as constables and beadles. They also elected an Alderman to represent them in the Common Council, which in turn elected the Mayor and Sheriff. The rise of guilds added a layer of complexity to this structure, as the guilds vied to elect their own as Alderman and ultimately, as Mayor.

The council governed tolls and the collection of taxes, as well as the assizes--which were essentially quality control boards for bread, wine, ale, and construction.
They also tested scales and governed weights and measures. There was also an assize for public nuisances. Disputes for each of the assizes were handled in courts known as Hustings, where for a fee a citizen could bring suit against another person for fraud, breach of contract, selling low-quality goods as high-quality, and the like. These courts also came to govern legal matters such as apprenticeship contracts, transfers of properties, and the like. These evolved into the municipal court system.

In England, all property technically devolved to the Crown, but in the case of cities, rather than the process of infeudation, the city's charter generally governed property in the city. As you might expect, this could lead to conflict between the city, who vigorously defended its rights to self-government, and the king, who would sometimes remove elected officials and appoint a warden to govern, such as during the baronial revolts of the 13th century, London sided with the barons (the future Edward I being so enraged by this that at the Battle of Lewes he chased the Londoners from the field of battle seeking vengeance--and lost the battle as a result) The Mayor of London was often considered to have the power of a king within the Square Mile of the city (inside the walls), with the actual King traditionally needing to ceremonially ask for permission to enter the city. The relationship between the King and London eventually mostly evolved into one of balancing powers, particularly after the rise of the guilds and the importance of London in governing international trade.

So the key for any city or town in England that wished to govern itself was having a Royal charter--which essentially gave the land and property of the city or town to the corporation of the city/town, allowing individuals to own or rent property there rather than from a noble holding it in fief. If a town did not have a charter, it was likely under the direct jurisdiction of what over local lord owned the land on which it stood. However, the nobility sometimes did own property within cities--although it was subject to the laws and statutes of that city just like any other property.

A good source for this is Caroline Barron's London in the Later Middle Ages: Government and People, 1200–1500.