A lord, the church, or the citizens, is the short answer.
We often think mostly about landholding lords in the medieval period as the linchpin of the "feudal" system, but the truth was far more chaotic. Landholding among the aristocracy was a complex and variable practice, with different customs prevailing in different regions, and the practice changing in those regions over time. A landholding lord might be the ancient holder of that piece of land, or might be the new owner through a grant or social promotion, or might be a renter, or a delegate, or appointee in lieu of the real owner. Sometimes, when the lord is absent, the lord's wife might control the land in the same way that he would, and she would wield all the same power and be expected to defend it in the same way, if necessary.
Landholding also wasn't national in the sense that we'd recognize it today. In places of longstanding conflict, like Normandy during the Angevin-Capetian dynastic conflicts, Norman-English landholders sometimes owned land and owed fealty, in some small ways, to Capetian-French chains of ownership. King John, at one point, even swore an oath as a vassal to the King of France in his capacity as a landholder in a certain disputed plot of land!
If in one's lordly holdings there was a city, that city would be in the charge of the lord. Cities were often fortified, either with their own walls and towers or huddled under the protection of a nearby castle. But if the city sat on lordly land, then the lord was in charge, though would often rule through appointed representatives who would work in various capacities for various purposes.
Rule by the church was similar. Land ownership was a major portion of church politics, and the church in general oversaw land in much the same way as any lay lord. They issued and collected taxes (or tithes), appointed officers and representatives to perform administrative work, collected the agricultural output, built or maintained fortifications, administered justice and punished criminals. While specific laws and customs might differ somewhat between a city or village administered by a lay lord or ecclesiastical lord, this was an expression of power built in parallel, rather than in competition. Market days and fairs and holidays would be celebrated in more or less the same way in an ecclesiastical landholding or a lay landholding, following the cultural customs of the people who lived there. It wasn't like an international border in the sense that we'd understand it today, even if the laws themselves might be somewhat different. There may be friction, sometimes, between who had the right to hold a fair on which patch of land, or who was able to collect the rents or output of a different patch of land, and sometimes this could be stupendously complicated, but this usually shook out at the level of regional politics instead of international wars. Sometimes.
The same was true of free cities. Free cities were especially common in the Holy Roman Empire, essentially being cities who had purchased or negotiated the right to their own political self-determination. They could structure their government how they wished, elect or appoint their own officers, collect rents and levy taxes and everything a lay or ecclesiastical lord could. Generally, free cities had some form of democratic structure, usually through the guilds that made up the city's production economy. Citizens had a voice in councils through their membership in a craft guild, and could be elected to office through their guilds, etc. Sometimes, this wasn't the case, and even neighboring free cities could be run very differently. Augsburg, for instance, had a strong guild culture, but neighboring Nuremberg was run through a council of patricians following a craftworkers rebellion in the 14th century.
To conclude, it depended on the city and on the time we're talking about. A city could be ruled by an absentee lord, the wife of a lord, or the appointed representative of a lord, either lay or ecclesiastical. It could be run by a council of citizens or a council of patricians, and administered by elected or appointed officials of that body. These powers all paralleled each other, and conflict over certain pieces of land or certain incomes of fairs or agricultural production often manifested in political levels, and could even lead to open feuds.