• The Episcopal Church (headquartered in the United States) established a presence in Europe during the 19th century in order to serve American Episcopalians who were spending significant amounts of time in Europe. The roots of the American Cathedral in Paris (pictured at left) extend back to a group of Americans who gathered for worship in the 1830s. They were given official membership in the Episcopal Church by the General Convention of 1859, when a canon was authored to admit churches in foreign lands. The canon encourages such outreach, while prohibiting the establishment of churches within the bounds of another Anglican jurisdiction. In 1971, the Convocation of American Churches in Europe was established. Today it is headed by Bishop-in-charge Pierre Whalon. Read more about its history here. (The 'America' in its title has now been changed to 'Episcopal' in order to recognize that TEC is not a "national" church.)
• The Diocese of Europe: In 1633, the growing number of Anglicans living outside of England were considered to be under the care of the Bishop of London. Groups of Anglicans developed around the world, including along the southern edge of Europe. In 1842, the Diocese of Gibraltar (a British territory) was formed to oversee the chaplaincies and congregations serving Anglicans in southern Europe. By 1833, the bishop of London added a suffragan (assisting) bishop to oversee Anglicans in northern and central Europe. By 1970, those areas were served by a single bishop, and in 1980, they were given status as the 44th diocese of the Church of England known as the Diocese of Europe.
• In 1880, an Episcopal bishop who had been appointed as Bishop to Mexico, helped establish the Spanish Episcopal Reformed Church. By 1894, a Spaniard was serving as bishop and by 1963, the SERC was in full communion with the Church of England. It was granted full membership in the Anglican Communion in 1980.
• The (Portugese Episcopal) Lusitanian Church, which formed in 1880 through a partnership between the Episcopal Church, the Church of Ireland, and local Roman Catholics, was also admitted to the Anglican Communion in 1980.
In addition to those Anglican bodies, the Churches in Europe have a rich ecumenical history, especially visible in the Porvoo Statement, which encompasses 11 Anglican and Nordic and Baltic Lutheran churches, including the Church of Denmark as who accepted the statment in December 2009. The Anglican Communion is also in full communion with the Old Catholic Churches of the Union of Utrecht.
From time to time, Anglican Europe's odd setup is singled out as an example of the Episcopal Church's participation in 'overlapping jurisdictions', and the existence of the Convocation of Episcopal Churches in Europe is used to make an argument that parallel Anglican jurisdictions could be seen as a legitimate ecclesiological structure. As a new fellowship of Christians claiming Anglican heritage (but not desiring membership in the Episcopal Church) finds a voice in North America, and as bishops from other Anglican provinces allow affiliation from American soil, the question of 'overlapping jurisdiction' comes into focus.
Christians have long held to the idea that there should be 'one bishop, one diocese', a maxim that traces back to Canon 8 of the Council of Nicaea (325): "for in one church there shall not be two bishops." (Canons 15 & 16 come into play as well.) So, the question goes, if Americans can violate this by having a bishop in Europe, why can't we have a bishop in America?' Such a characterization oversimplifies the history of Anglican jurisdiction in Europe. There was no existing Anglican diocese in Europe* when TEC or the CofE sent bishops to provide oversight for their nationals living there. The 'overlapping' jurisdictions grew up alongside each other organically rather than as the result of any division or intentional competition. As the world becomes smaller, though, these European Anglican bodies have begun a conversation about how they might go forward together. The European Consultation and European Partners in Mission dialogued from 1998 to 2004, when they issued a statement saying that while a unified structure was not yet workable, the development of deeper collegiality and networked ministries was needed and encouraged.
*Why Anglicanism was not represented in continental Europe would mean writing a post on the various streams of reformation that altered the course and structure of Western Christianity. I'd like to address that, but for now Wikipedia has this fairly helpful article on 'the Protestant Reformation'.