This is an incredibly broad question, and the answer will vary widely depending on time and place. The Roman Empire was governed differently from, say, the Frankish kingdom, and also differently depending on what period of Rome you are speaking of. Is there a specific time period or country you are interested in?

I will give you one example, from my area of expertise, of how a pre-modern empire/kingdom could be governed. This is, of course, a rather clean summary, lacking in detail and realistic messiness but should give you some idea of the complexities and methods of rule.

The Carolingian Empire under Charlemagne and Louis the Pious (c.750-c.840) consisted of modern day France, the Netherlands, Bavaria, northern Italy, much of Germany and a chunk of northern Spain.

The king, or post 800 (when Charlemagne was crowned emperor in Rome) emperor was the ultimate ruler of all of this territory. Under the king were the counts, abbots, and archbishops/bishops responsible for administering over counties and episcopal sees respectively. I list both because administration of the realm was a joint effort between both secular and ecclesiastical figures (who generally came from the same class of aristocratic families). This makes it sound very well organized, and in some ways it was, but the empire was big, and was, in fact, an amalgamation of several smaller realms. So different areas could function in different ways. In some areas (especially borders) a count might hold multiple counties and function as a Margrave (marcher lord). In some areas a royal monastery run by an abbot might exercise major authority. In other areas an archbishop might be the major administrative figure. On a local level counts, abbots, and bishops (who were typically from families that enjoyed dominance in the area) oversaw justice, made sure laws and royal edicts were enforced, raised levies for war/defense, oversaw the spiritual wellbeing of the people etc. Below these figures were a whole array of other shadowy people ensuring the smooth functioning of the area.

In addition to this infrastructure, the Carolingians also had officials known as missi, literally "sent ones". These figures, who were themselves counts and bishops, served the king by traveling to an area (called a missatica) usually in pairs (one Bishop and one Count) to basically audit the area, spread new laws and rules and generally ensure that things were being run correctly. Missi could also be sent on specific tasks, internally and as envoys to foreign lands. In addition, certain royal officials (the chancellor for instance) took care of duties both in the court and realm wide (in some ways like the president's cabinet in modern America, though we don't want to stretch this metaphor too tight).

Each year a general assembly (sometimes multiple assemblies, especially later) would be held. Early in the realm these often occurred just before a major campaign into, say, Saxony. Later they were typically held at one of the main royal palaces dotted around the realm. Aristocrats and others would come to these assemblies and discussion would be held on major topics. The king would, with a close circle of advisors, likely set the agenda and also meet in smaller groups. Major business would be taken care of (direct requests for royal adjudication, questions about specific needs of specific areas, the granting of land or titles, meetings with foreign envoys etc.). Often at the close of these assemblies documents, called capitularies would be issued with new rules and decisions to be applied realm-wide in the future. These assemblies were also a time to show patronage to people, share news/gossip, feast and publicly display the king's largess and authority.

Throughout the year other events, such as episcopal councils, the issuing of immunities, charters and other documents (basically designed to give property and rights to institutions, or affirm a royal decision) would also occur. The king had a body of clerics responsible for issuing documents.

What you end up with is a complex system where the king has final say, but rules with consensus by giving out authority and favor to local interests, while also cultivating a semi-bureaucratic system of administration. Patronage and favor are very important, power trickles down from king to counts to local lords etc. King's have to be careful not to anger too many people or they might find themselves facing rebellion, but counts equally have to be sure to behave because the titles they hold (and the lands that go with them) are revocable.

This leaves out a lot of stuff (economic transactions, taxation and tolls, etc.) It is really in some ways quite complex while still being incredibly "under organized" by modern bureaucratic standards.

Some books to check out if you have further interest in Carolingian (or early medieval) governance.

The Carolingian World a really solid modern textbook on everything Carolingian.

McKitterick's Charlemagne a nice analysis of how an emperor rules and builds an empire.

Dutton's Carolingian Civilization a reader filled with primary sources, including a bunch of governmental documents.

Anyway, I hope that gives you some idea about how one particular medieval kingdom in the Early Middle Ages was governed. Happy Reading.