At the lowest level, there is the “tithing,” made up of every male citizen who is a feudal tenant to a lord or manor (or “villein”) between the ages of 12 and 60; traditionally, as befit the name, a tithing was meant to be made up of ten villeins — however, it tended to be all those living in a hamlet or on the same street in a village, so that number varied. Every member of the tithing made a pledge to observe and uphold the law. Each was expected to report a crime occurring within his tithing and to deliver the criminal to the local law-enforcement officer (from smallest to largest area of administration, these were the steward of the lord, the bailiff, the reeve, the mayor, the constable, the bailiff of the hundred, and finally the sheriff). Failure to do so could incur a heavy fine. The head of the tithing, known as the “capital pledge,” made sure that those beneath him were upholding the law and reported crimes to the courts.

The lowest level of court was the manor or borough court, which encompassed just the land held by a lord or manor. The rules for this court could vary from place to place. It generally oversaw any business relating to the safe and efficient running of the manor. Cases here involved basic managerial aspects of the manor, such as land maintenance, cattle management, and the infringement of lord’s rights. Minor issues involving villeins, like fist fights and slander, could be dealt with here. Small fine were also levied, sometimes for “moral offenses.”

The next level of the courts took place at the county level and was overseen by the Sheriff. This court would have been a very, very rough equivalent of modern day city court and small claims court. It mostly oversaw routine business, like swearing in of officials or sharing of royal proclamations, and very small civil claims. It was also the place where outlaws were declared (which more or less served as a summons to court, differing from a modern day summons notably in that the fifth summons with a failure to comply resulted in an automatic penalty of “beheading on sight”). County court also acted as a Court of Original Jurisdiction by taking on preliminary hearing of Crown pleas and referring those that qualified to royal courts.

The next level of courts was referred to as “the Hundred Court,” which met once a month. (A “hundred” was the division of the shire for military and judicial purposes — you might think of it as the difference between the city and the county.) Headed by a bailiff and judged by a jury of twelve freemen (men who, unlike villeins, were not beholden to a lord or manor), this court acted in a manner similar to federal courts today. When issues arose between villeins of different manors, rather than being dealt with at the Sheriff’s level, the issue was escalated to the court overseeing the larger area. Also, freemen — who cannot be held to account at the manorial level because they are not beholden to a local manor or lord — were judged for criminal offenses at the Hundred Court.

Twice a year, the sheriff’s court came to the Hundred Court. On these occasions, royal court referrals were made, felons were taken into custody, and large fines were levied against tithings that failed to report crimes.

Next were the royal courts, which included the Court of the Exchequer, the Court of the King’s Bench, and the Court of Common Pleas. The King’s Bench and Court of Common pleas visited each county, usually twice a year, to hear important criminal cases and appeals from lesser courts, including those involving financial matters. Criminal cases were heard by a jury, and almost always resulted in either a death penalty by hanging or a verdict of innocence and release. Sentences of life imprisonment existed but were very rare. Some crimes could be punished with loss of limb, though this was also less frequent.

Because Sheriffs were not given the power to try serious criminal cases, eventually these royal courts found they had more work to do than could be done. As a result, the office of the Justice of the Peace evolved, and was officially established by Kind Edward III in 1361. These Justices — also known as magistrates — were appointed by the crown and given the power to arrest, bail, and punish “breakers of the peace.” Generally they were men “learn’d of the law”, with three or four being appointed per county for the purpose of lessening the burden on the royal courts.

Sources:

Bennett, Life on the English Manor
Harding, Law Courts
Summerson, Structure of Law-Enforcement

I think it would be particularly helpful to try and narrow this down, since medieval period is a long period, and even if we limit this to Europe, we still get plenty of substantially different legal systems, with of course developing ecclesiastical courts. Barring clarification, for example in the Holy Roman Empire, we can roughly characterize to manorial courts, town courts, guild courts and ecclesiastical courts. How these interact and develop would of course depend on time and place in question, as do other jurisdictional bounderies between them, whether these courts had the ability to met out certain punishment or not, the ability of appeal to provincial courts ( with the development and centralization of provincial bureaucracy, and which privileges the province had, as certain duchies and states had, like privilegium de non appellando, where the highest provincial court was the final appelate court, except for imperial princes or other imperial titles. A commoner might appeal the vedict of a manorial court, but provincial courts ( where a provincial lord, but later a magistrate, sometimes noted as iudex provincialis, appointed by him, would preside ) would rarely take it up, except in rare cases - in early modern period, provincial chambers delegated to this would usually hear complaints from servants, under incentive to try avoid peasant uprisings, although most studies have rather showcased the ineffectiveness of these bodies. Provincial courts also had jurisdiction for harsher criminal offenses, and per some town statutes ( and depending on a city ) to hear cases typically under town´s court jurisdiction, if inexcusable deficiency was identified, like iustitia denegata vel protracta.

Provincial bodies in matters of nobility went from seperate Hofrechten and Landrechten courts into one body, Landschranne, although the intial distinction still remained in relation to law, the former being for disputed of possessions, the latter for other disputes - even here though, provincial lords would delegate this to the magistrates, typically Landeshauptmann, who would preside over such proceedings. But as stated, depending on the privileges of the province, state, etc. and depending on the nobles in question, they might have the appelate option to imperial court, or later in early modern period with subsequent development, to other bodies.

I might of course be able to say much more if the initial reservations were clarified.