Some famous exclave
Alaska: Territory of the United States, surrounded by Canada and the Baring Sea. Acquired off Russia after Canada's founding. State in its own right.
Kaliningrad: Territory of the Russian Federation, surrounding by Poland, Lithuania and the Baltic Sea. Formerly part of East Prussia and remained in Soviet hands after the Second World War. Boris Yeltsin had promised to hand Kaliningrad to Poland but he changed his mind after Poland sought entry into NATO. Oblast in its own right.
Oecussi-Ambeno: Territory of Timor Leste (East Timor), surrounded by Indonesia and the Savu Sea. Oecussi was settled by the Portugese in the sixteenth century but they later also settled in East Timor, eventually moving the administrative capital to Dili. They never got around to agreeing with the Dutch to connect both Portugese colonies. Under Indonesian occupation (1975-1999) the enclave was under the same special status jurisdiction as East Timor.
Cabinda: Territory of Angola, surrounded by the Democratic Republic of the Congo and the Atlantic Ocean. Originally attached to Angola, until the 1885 Berlin conference decided to extend the Congo Free State's borders to the mouth of the Congo, and Cabinda was isolated. Province in its own right.
Temburong: Territory of Brunei surrounded by Malaysia and the Brunei Bay. Daerah (district) in its own right.
Northwest Angle: Part of the county of Minnesota in the United States, it is on a peninsula extending eastwards from Canada into the Lake of the Woods. It is at the eastmost part of the US/Canadian border where the frontier stops being straight, and instead wiggles at the start of the Great Lakes districts, as per the 1783 Treaty of Paris. An 1842 treaty between Britain and the United States, in configuring a lateral line along a triangular shaped geographical feature, gave the United States the northwest portion of the Lake of the Woods.
Point Roberts: Part of the county of Whatcom of Washington state in the United States, it is on the southern portion of an island that is shared with Canada.
Ceuta and Melilla: Territories of Spain, surrounded by Morocco and the Mediterranean Sea. Captured by the Portugese in 1445, Portugal yielded Ceuta to Spain on January 1, 1668, at the signing of a peace treaty at Lisbon mediated by Charles II. Melilla was the frontier of the Kingdom of Tremecén and the Kingdom of Fez, when Spain captured the city in 1497. Both are autonomous cities in their own right.
Nagorno-Karabach: Territory of Armenia surrounded by Azerbaijan. Variously occupied by Turks, Arabs, Azeris and Persians. Was to be returned to Armenia by the Bolsheviks in 1920, except they reneged in order to buy Azeri support in an attempt to export Communism to Turkey. Remained separated throughout the lifespan of the Soviet Union, and since then has been a source of conflict between Azerbaijan and Armenia. Country in its own right - except that no other country except Armenia recognises its own independence. There are also three Azeri exclaves within Armenia, the largest being Nakhchivan.
The Gaza Strip: Territory of the Palestinian Authority surrounded by Israel, Egypt and the Mediterranean Sea.
Castel-Gandolfo: Territory of the Holy See surrounded by Italy. The Pope's summer residence. In addition thirteen buildings in Rome enjoy extraterritorial rights.
And other lesser known exclaves:
Baarle-Nassau: Territories of Belgium surrounded by the Netherlands, with a few counter-enclaves. See my separate node.
Büsingen: Territory of Germany surrounded by Switzerland
Barak: Territory of Kyrgyzstan surrounded by Uzbekistan. There are several small Uzbek exclaves within Kyrgyzstan
Campione: Territory of Italy surrounded by Switzerland
Llivia: Territory of Spain surrounded by France
San'kovo-Medvezh'e: Territory of Russia (consisting of two villages) surrounded by Belarus.
Mahda: Territory of Oman surrounded by the United Arab Emirates. But inside Mahda is a territory of the United Arab Emirates surrounded by Oman.
The Cooch Behar enclaves: Found along the Bengali-Indian border. Within India there are 71 Bangladeshi enclaves, of which three contain an Indian counter-enclave. Within Bangadesh there are 102 Indian enclaves, of which 21 contain a Banglaeshi counter-enclave (and one of which contains a seven square metre Indian counter-counter-enclave). The complexities originated around the time of the Mughal Empire in the early eighteenth century, and became curlier after Partition. Suffice to say, despite all the antagonism between New Delhi, Karachi and later Dacca, the people in the Cooch Behar all live quite harmoniously with each other.
One can also question if the dozen or so colonies around the world (Gibraltar, French Guyana etc) are exclaves depending on how one defines the amount of autonomy it exercises, and how much of a foreign territory an exclave should be surrounded by.