The UN was established in October 1945, after World War II as an international organization facilitating cooperation in, among other areas, international law. It was a replacement for the inefficient League of Nations around 1945. The United Nations suceeded the League of Nations, a similair international organization formed in 1919 to work for peace following the First World War.
Based on observation of the events in Aliens: Steel Egg, the United Nations became strengthened beyond its original intent over the decades, as the UNIC controlled most of the territories in the Solar System including Earth. Thus, there is reason to believe that the strengthened form of the UN functioned as the highest government body on Earth and many off-world human colonies at the time of Aliens: Steel Egg.