The International Confederation of Wizards is the wizarding version of the United Nations. British seats on the International Confederation of Wizards are probably selected by the Minister for Magic, possibly subject to the approval of the Wizengamot.
The International Confederation of Wizards is headed by a person who holds the title "Supreme Mugwump". The first appointed Supreme Mugwump was Pierre Bonaccord. However his appointment was contested by Warlocks in Liechtenstein, due to his opinions on stopping Troll-hunting and rights for Trolls. Liechtenstein, who had been dealing with several nasty Troll communities at the time refused to attend the first conference, causing trouble for the new institution.
In 1692, the Confederation instituted the International Statute of Wizarding Secrecy to hide the wizarding world from the Muggle world. After weeks of discussion, they decided on how the wizarding world would retreat into hiding, including the covering up of various magical beasts and eliminating them from the Muggle consciousness.
Albus Dumbledore was appointed Supreme Mugwump but was dismissed from his position following the Ministry's denunciation of Voldemort's return. He was restored as a member after the Ministry's acceptance of Voldemorts return but not as Supreme Mugwump.