Typically it seems all us cases have a judge. Theres also a lot of different types of courts. Are all the judges at the same job level? Do they have managers? Who are they?
Do they performance metrics they have to meet? What are those metrics?
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Most particular courts have a chief judge, a position that is paid slightly more than that of a usual judge for a huge additional burden of administrative work managing the court house (e.g. courtroom assignments, policies on implementing case assignments, overseeing the jury system, security measures, hiring court clerk's office and court building staff, reviewing court budgets, deciding whether to close the courthouse for weather or pandemics and the like, etc.).
In Colorado, there is a state appointed panel for each court that in connection with an outside private contractor evaluates judges periodically and prepares a report recommending or not recommending their retention in periodic retention elections, and there is a separate appointed panel that investigates allegations of judicial misconduct.
In many other states, some or all judges have to run for re-election when their terms end and the voters decide if they should stay in office. Sometimes judges in these states are nominated by the appropriate level of one of the major political parties.
In the federal courts, since judges serve for life and there is far less discretion to discipline them, Congress is the main supervisor in the sense of having the power to "fire" a judge.
Trial court judge's substantive legal decisions are subject to review by courts with appellate authority over their court on a case by case basis. For courts of general jurisdiction, direct review of their cases on appeal is vested in an intermediate court of appeal in all but a few small states. For courts of limited jurisdiction that handle smaller civil disputes and/or misdemeanor criminal cases, direct appellate review is usually vested in the courts of general jurisdiction above it.
In each state, the state judiciary as a whole has at its pinnacle a highest court, usually called the state supreme court, although a few states call it something else.
It is the final court of review in the appellate process.
It also has the authority to intervene in particular judicial decisions that would not ordinarily be ripe to raise in an ordinary direct appeal (e.g. questions about whether allegedly attorney-client privileged or other privileged evidence must be disclosed), including the authority, in rare cases to order that a judge in a case that was appealed be removed from a case. The federal court system and some larger states vest this authority in intermediate appellate courts as well.
State supreme courts also have general administrative authority over the judiciary as a whole. The state supreme court (primarily through the chief justice of the state supreme court) in its administrative capacity handles budget issues, formulates court rules, prepares annual reports on the judicial branch, lobbies for state funding, is responsible for auditing lower court finances, makes IT decisions for the state judiciary as a whole, and makes myriad other policy decisions, although it does not directly supervise judges in the way that a line manager in a government agency would supervise that manager's subordinates.
The closest thing a judge has to a manager is the Chief Justice of their court. They can set internal practices and expectations, and have discretion over assignments. See Peter W. Hogg, "The Role of a Chief Justice in Canada" (1993):
the Chief Justice of each superior court has the power to assign judges to particular cases. This includes the power to assign himself or herself to a particular case. The Chief Justice also controls the list of cases, including the order in which they are to be tried or heard and the courtrooms in which they are to be tried or heard.
How this power is exercised is generally not public. A rare public dispute about this assignment process can be read in the competing reasons in R. v. Gashikanyi, 2017 ABCA 194, with one justice emphasizing the "inherent risk flowing from such non-random assignments" and another emphasizing that there are "legitimate considerations," including expertise, that justify preserving a chief justice's discretion to assign judges to panels.
These decisions might in practice be largely delegated to court administration or to other positions within the judiciary (see Canadian Judicial Council, "Alternative Models of Court Administration" (2006)). E.g. At the Federal Court (at least in 1999), "[t]he Associate Chief Justice's Office assigns judges and prothonotaries to hearings and fixes the timetables of those persons" (Canada Post Corp. v. Varma, 1999 CanLII 8426).
There is no almost no publically available information about internal performance metrics that judges may be expected to meet. However, the Canadian Judicial Council has resolved that judgments should be delivered within six months after hearings, except in special circumstances (referred to in this Notice to the Profession from the Court of King's Bench of Alberta; also codified at Art. 324 of the Code of Civil Procedure of Québec). Some courts report on their ability to meet this target (e.g. Court of Appeal for British Columbia Annual Report 2020, at p. 25).
The appeal process is another method of "supervising" the work of judges. A judge's work is reviewable for error by courts of appeal.
Another kind of "management" is the oversight role of the Canadian Judicial Council. The CJC has the power to make inquires and the investigation of complaints or allegations concerning the capacity of a judge to execute the functions of their office for reasons of age or infirmity, misconduct, failure to execute their functions, or being in a position incompatible with the execution of those functions (Judges Act, ss. 60-65). The outcome of this investigation can be that the CJC recommends to the Minister of Justice that the judge be removed from office (Government of Canada, "The Judiciary").
All of the above is about judges within the judiciary. Tribunals are somewhat court-like bodies established within the executive branch of the government to adjudicate certain disputes, like entitlement to benefits, remedies under Human Rights Codes, etc. The tribunal members may very well have something more akin to a traditional "manager" with performance targets and reviews. The statute creating such a tribunal can establish whatever supervisory mechanism it desires (Ocean Port Hotel Ltd. v. British Columbia (General Manager, Liquor Control and Licensing Branch), 2001 SCC 52):
It is well established that, absent constitutional constraints, the degree of independence required of a particular government decision maker or tribunal is determined by its enabling statute. It is the legislature or Parliament that determines the degree of independence required of tribunal members. The statute must be construed as a whole to determine the degree of independence the legislature intended. ... like all principles of natural justice, the degree of independence required of tribunal members may be ousted by express statutory language or necessary implication. ... Ultimately, it is Parliament or the legislature that determines the nature of a tribunal’s relationship to the executive.
One example are the "full board meetings" used by the Ontario Labour Relations Board "to foster coherence and maintain a high level of quality in the decisions of the Board" (Iwa v. Consolidated-Bathurst Packaging Ltd.,  1 SCR 282).