The Union of European Football Associations (UEFA) is the European football, futsal and beach soccer governing body, although there are a number of Member States mostly or completely in Asia. It is the FIFA world's governing body, one of six continental confederations. UEFA is made up of 55 members of the national association.
UEFA serves as the European national football body, runs the country and club competitions including, and tracks prize money, rules and media rights for these competitions, including the UEFA European Championships, UEFA League, UEFA Europa League, the UEFA Europa Conference League, the UEFA Super Cup.
The first secretary general was Henri Delaunay and the first president was Ebbe Schwartz. The new president is Aleksander Čeferin, the former president of the Slovenian Football Association. In september 2016 he was chosen as UEFA's seventh president at the 12th Extraordinary UEFA conference at Athens.
Why some clubs are allowed to participate in competitions outside of their territory?
UEFA consists of 55 members of national organisations. Some countries, including Gibraltar and the Faroe Islands, are not even Sovereign countries, although sovereign countries like Monaco and the Vatican City are not represented. Some people, such as Kazakhstan, are not accepted in Europe geographically.
The National Body, which now appears to no longer manage the top domestic competitions, runs national cups. Often secondary cups are also open, for example a league cup. Cups are also only eligible for non-professional clubs outside the Top Service(s) or provincial cups. However, eligibility to play is as closely related as many cups are.
For example in Lichtenstein, seven teams play in the Swiss League system, but they have their home cup competition which gives clubs from one of the two countries that are twice locked up in the world an opportunity for their national flag to compete in the Europa League. That's not the same as AS Monaco, though.
Monaco is an autonomous country but has no competitive domestic league or cup because of its population. The club is also part of the French system and represents France in Europe. Cardiff City, Swansea City and Newport County take part in the English Leagues even though they are from Wales. Another case is Derry City, founded in Northern Ireland, who participate in the Irish League.