Is it like our "Tribunal de Police" for speeding on the road, noises
during the night and others petty things of the same kind?
Or is it more like "Tribunal correctionnel", when it covers matters
which are a little more important?
In some cases the former, in some cases the latter, and in some cases more than either of the courts, as they would also have authority over some civil lawsuits and administrative law matters as well. Practice has varied over time and from one state to another (and even within individual states).
Justice courts and police courts are terms used for something similar to the Tribunal de Police, but sometimes a bit more serious and often with jurisdiction over very small lawsuits in which lawyers are usually not involved called "small claims" cases colloquially.
While many U.S. municipalities have administrative style tribunals limited solely to traffic offenses and/or solely limited to parking offenses, there are few limited jurisdiction courts in the U.S. with jurisdiction identical to a "Tribunal de Police" in France. Most would have jurisdiction over more serious crimes or quasi-criminal offenses and many would have some limited civil lawsuit jurisdiction.
In practice, however, many (maybe even most) municipal courts in the U.S. handle a similar mix of cases to a "Tribunal de Police" even though it has jurisdiction over more serious cases and over lawsuits, as a matter of the customary division of authority between a city attorney or police officer who prosecutes cases in municipal court (Rhode Island, for example, has non-lawyer police officers prosecute many minor criminal offenses in limited jurisdiction courts), and a county or state level district attorney who prosecuted more serious cases in a court at the county level that is part of the state court system. Many municipalities maintain courts of their own primarily to raise traffic fine revenue that would otherwise go to a county or state government if prosecuted in their courts.
Friedman, in the excerpt quoted, is referring collectively to all courts of limited jurisdiction, some of which are true municipal courts, and some of which are lower courts in a larger state unified court system.
In General
Municipal courts in the U.S. are creatures of state law (and sometimes are outside the state court system).
All municipal courts are courts of limited jurisdiction as opposed to courts of general jurisdiction that can consider the most serious criminal charges and all manner of civil claims.
No municipal courts have jurisdiction over matters exclusively in the jurisdiction of the federal courts (e.g. enforcing federal criminal laws and lawsuits involving intellectual property disputes).
It would be very unusual for a municipal court to have jurisdiction over felony trials, but often they would have authority to issue search warrants, arrest warrants, and arraign defendants (i.e. formally charge, take an initial plea from a defendant, set bail for a defendant, and assign a lawyer to an indigent defendant) for felony charges that would then have further proceedings post-arraignment in a higher court.
It would be very rare for municipal courts to have civil jurisdiction over title to real property or marital status or child custody, and it would be unusual for municipal courts to have jurisdiction to provide most kinds of injunctive relief (with the common exception of temporary protective orders in the face of threats of physical harm).
Since it frequently comes up in other questions, it is also worth noting that in the U.S. municipal courts generally have jurisdiction to declare statutes that they are asked to apply or interpret to be unconstitutional and to adjudicate the constitutionality of actions of state officials in the context of criminal cases.
Municipal courts almost always (but not 100% of the time) have jurisdiction only over cases arising in the territory of the municipality which it serves.
In New England, all territory is part of some town government which handles many functions (but not general jurisdiction courts) that county governments would handle elsewhere.
Outside New England, there are many states where lots of territory is outside the jurisdiction of any town, township, or city, and has local government functions otherwise provided at those levels of government provided at the county level or by special districts with limited functions (e.g. a school district, or a water and sewer system district).
In the Midwest, there are many state with formal township government that covers the entire state, but usually, townships, unlike municipalities called towns, don't have law enforcement agencies or courts of their own (with their ordinances enforced in state courts of limited jurisdiction at the county level by county officials outside of a municipality), and instead handle only minor infrastructure maintenance of local roads and cemeteries and sometimes land use issues in areas outside an incorporated city or town.
The Colorado Example
In Colorado, a municipal court's jurisdiction is limited to ordinance violations (i.e. violations of laws enacted by the municipality itself) taking place in the municipality, which can range from parking infractions to zoning code violations to minor misdemeanors that are also state crimes. Typically, those offenses have punishments up to twelve months in jail and fine of up to $1,000 (U.S.). Colorado's municipal courts do not usually have jurisdiction over civil lawsuits seeking money damages, but a small number of municipal courts in Colorado have been granted civil jurisdiction in certain ordinance cases, such as cases involving land use, under municipal home rule powers, in addition to quasi-criminal jurisdiction.
Most of the territory in Colorado is not in any municipality or town, and Colorado has no township governments (the term "township is also a term used to describe areas of about 36 square miles for purposes of legal descriptions of real property pursuant to an official surveying system).
Municipal court judges in Colorado, are appointed by municipal officials, are regulated by the attorney regulation officials rather than judicial regulation officials, and often serve as a municipal judge in more than one municipality. Offenses in municipal court in Colorado are prosecuted by municipal officials rather than the state district attorney.
Municipal courts in Colorado are only sometimes courts of record (i.e. courts where appeals are based upon what happened in municipal court rather than via a new trial from scratch in a higher court). They are not administratively part of the state court system.
County courts in Colorado are courts of limited civil jurisdiction (up to $25,000) that can also handle state law misdemeanors and arraignments for felonies and are courts of record with lawyer-judges (except for four exceptions out of more than 100 judges in a few rural counties) and are part of the state court system. Outside of municipalities in Colorado, county ordinances are enforced by county officials in county courts.
In Colorado, courts of general jurisdiction are called "District Courts". Colorado also has a probate court, and a juvenile court in the City and County of Denver which handle a portion of the cases handled by District Courts elsewhere in the state.
The New York State Example
In contrast, in New York State, municipal courts in different municipalities have different jurisdictions, some substantial and integrated into the state unified court system, and some minor, with non-lawyer judges often presiding in Justice Courts. In New York State:
City courts handle the arraignment of felonies, try misdemeanors and
lesser offenses, and try civil lawsuits involving claims of up to
$15,000.
The New York City Criminal Court and the New York City Civil
Court are the local courts in the 5 boroughs of New York City.
(Ed. The New York City Civil Court has jurisdiction over civil lawsuits
involving up to $25,000.)
Some city courts have small claims parts for the informal disposition
of matters involving claims of up to $5,000 and/or housing parts to
handle landlord-tenant matters and housing violations.
The District Court is the local criminal and civil court in Nassau
County and the five western towns of Suffolk County. It arraigns
felonies and tries misdemeanors and lesser offenses, as well as civil
lawsuits involving claims of up to $15,000, small claims and small
commercial claims up to $5000, and landlord-tenant actions.
Justice courts (town and village courts) try misdemeanors and lesser
offenses in towns and villages. These courts are the starting point
for all criminal cases outside cities, and handle a variety of other
matters including small claims, traffic ticket cases and local zoning
matters. They also arraign defendants accused of felonies. These
courts may hear civil lawsuits involving claims of up to $3,000
(including small claims cases of up to $3,000). Unlike all other
courts which are state-funded, the town and village justice courts are
locally funded. Justices are chosen by local election and are not
required to have a law degree, license to practice or any other formal
legal or law enforcement training, although upon election they are
provided with training and are subject to an annual continuing
education requirement.
The Traffic Violations Bureau (TVB) is a part of the state Department
of Motor Vehicles that adjudicates non-criminal traffic violations
(other than parking violations) in New York City.
The New York City Office of Administrative Trials and Hearings (OATH)
is an agency of the New York city government that conducts
administrative hearings, overseeing the operations of four tribunals:
the OATH Tribunal, the Environmental Control Board, the Health
Tribunal, and the Taxi & Limousine Tribunal.
Other U.S. court terminology is also inconsistent
The terms "county court" and "district court" sometimes are used in particular court systems to refer to limited jurisdiction courts and are sometimes used in particular court systems to refer to general jurisdiction courts.
The term "circuit court" is sometimes used to refer to general jurisdiction trial courts, and is sometimes used to refer to appellate courts.
The term "supreme court" is sometimes used to refer to the highest appellate court in a court system, sometimes used to refer to the general jurisdiction trial court in a system, and sometimes use to refer to an intermediate jurisdiction appellate court.
At least when I was practicing actively in New York State, the "Family Court" didn't have jurisdiction over divorces which was handled in the "Supreme Court" a general jurisdiction trial court which was also an intermediate appellate court.
The term "superior court" is usually used for a county level trial court with jurisdiction over some or all felony cases but not jurisdiction over civil lawsuits.
Courts with probate court jurisdiction have a variety of amusing names, including "orphan's court" and "surrogate's court" and "probate court" and "chancery court" when they are not part of another court.
Some courts of general jurisdiction also have jurisdiction over minor criminal and civil cases and customarily delegate authority over the smaller cases to junior judges appointed by the senior judges who are called magistrates. The term "magistrate" is also applied in some legal contexts (especially constitutional criminal procedure) in a manner that includes judges of limited jurisdiction courts that aren't part of a general jurisdiction court even though they are subordinate to the general jurisdiction court.
