Amtsgericht is German for Local District Court, situated in Germany

Germany

Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...

 in almost every larger capital of a rural district.
It mainly acts in Civil and Criminal law affairs. It forms the lowest level of the so-called ordinary jurisdiction of the German judiciary (German Ordentliche Gerichtsbarkeit), which is responsible for most criminal and civil judicial matters. The next level is called Landgericht.
The German Amtsgericht may be compared to the British Magistrates' Court.
The main areas of an Amtsgericht's jurisdiction

Jurisdiction

Jurisdiction is the practical authority granted to a formally constituted legal body or to a political leader to deal with and make pronouncements on legal matters and, by implication, to administer justice within a defined area of responsibility...

 are:

• court of first instance for civil case where the subject of litigation is worth 5,000 euro
  Euro
  The euro is the official currency of the eurozone: 17 of the 27 member states of the European Union. It is also the currency used by the Institutions of the European Union. The eurozone consists of Austria, Belgium, Cyprus, Estonia, Finland, France, Germany, Greece, Ireland, Italy, Luxembourg,...
  
  s or less, and for litigation involving rental agreements, marriages, alimony and child custody.
• court of first instance for criminal cases where a fine or a prison term of no more than four years can be imposed.
• administration of several public register such as the companies', the associations', the cooperatives' and the land owership register.