An exclave is strip of land that belongs to an entity (like a country or a region) but that is not connected to it by land (islands are not counted). The strip of land is surrounded by other political entities. A good example is Kaliningrad Oblast (the region around the Russian city Kaliningrad). It belongs to the Russian Federation, but is separated from the rest of that country by territory belonging to Lithuania and Poland.

Although both meanings are close, an exclave might not be an enclave.  An enclave is completely surrounded by one other country.   For example, Kaliningrad is surrounded not by one state, but by two: Lithuania and Poland. It also borders the Baltic Sea. It is not an enclave.  On the other hand, the Spanish exclave of Llivia is an enclave in France.

Many exclaves today have some of idea of becoming independent, especially if the exclave is far away from the mainland.

A much more obscure use means, in medical discussion, a detached part of an organ, as of the pancreas, thyroid, or other gland.

The lists below are of various types of exclaves that are not enclaves.

True exclaves
 Angola: Cabinda
 Azerbaijan: Nakhichevan
 Belgium: Baarle-Hertog
 Brunei: Temburong
 Croatia: Dubrovnik
 East Timor: Oecussi-Ambeno
 Germany: Büsingen am Hochrhein
 Italy: Campione
 Netherlands: Baarle-Nassau (parts)
 Oman: Musandam
 Russia: Kaliningrad 
 Spain: Llivia
 Northern Cyprus: Kokkina (de facto).
 United States: Alaska

Subnational exclaves
Exclaves of administrative divisions below the national level.

Many of the cantons of Switzerland have exclaves:
Bern has two small exclaves, Münchenwiler and Clavaleyres; the first is entirely surrounded by Fribourg, the second by Fribourg and Vaud's exclave.
Lucerne has two pieces separated from the main territory by the Vierwaldstättersee; one borders Schwyz, the other Nidwalden.
The "half-canton" of Obwalden is in two large pieces, separated by a strip of Nidwalden.
Nidwalden's district of Hergiswil is separated by an arm of the Vierwaldstättersee.
Fribourg is a large exclave bounded by Vaud and Lake Neuchâtel; two enclaves, Surpierre and Vuissens, within Vaud; and another enclave, Wallenbuch, within Bern.
Solothurn has two exclaves, Dorneck and Thierstein, both bounded on the south by Basel-Country and on the north by France; as well as the enclave of Steinhof which is entirely surrounded by Bern.
Schaffhausen, the only canton lying mostly on the north bank of the Rhine, is cut into three parts by German territory that links Germany to the Rhine.  The middle part is the largest, and surrounds the German enclave of Büsingen.  The upper and middle parts border on Thurgau, the middle and lower parts on Zürich.
The divided canton of Appenzell is surrounded by Sankt-Gallen.  Since its partition on religious lines in 1597, Appenzell Innerrhoden has three small exclaves, one enclave in Appenzell Ausserrhoden and two bordering also on Sankt-Gallen.
Vaud has one exclave, Avenches, bordered by Lake Neuchâtel, Fribourg and the tiny Bernese exclave of Clavaleyres.  The coast of Lake Neuchâtel is thus in seven pieces belonging to four cantons: clockwise from the north they are Neuchâtel, Bern (main), Vaud (Avenches exclave), Fribourg (main), Vaud (main), Fribourg (exclave), Vaud (main).
Geneva has two small exclaves within Vaud, together making the commune of Céligny.  The larger has a shoreline on Lake Geneva.
The secession of the new canton of Jura in 1979 left Bern temporarily with an exclave bounded by Jura, Solothurn (main and both exclaves), Basel-Country and France.  The exclave, Laufental, joined Basel-Country in 1994.

United States:
Kentucky: The westernmost part of the state, a piece of land known as the Kentucky Bend, is surrounded on three sides by the Mississippi River, with Missouri on the opposite shore. Road access to the rest of Kentucky is only available by first going south into Tennessee.  This exclave exists because the Mississippi, which forms the boundary between Missouri (right bank) and Kentucky/Tennessee (left bank), crosses latitude 36°30', which is part of the border between Kentucky and Tennessee, three times.
Louisiana: St. Martin Parish has an exclave which is separated from the main part of the parish by Iberia Parish.
Massachusetts: Norfolk County is not in one piece.  The towns of Brookline and Cohasset are part of Norfolk County, but are surrounded on all sides by other counties: Brookline by Suffolk and Middlesex, Cohasset by Plymouth.
California: The City and County of San Francisco has an exclave on Alameda Island, across San Francisco Bay and adjoining Alameda County. The exclave came into being as land on the border between the two counties was reclaimed from the Bay to build the Naval Air Station Alameda, now decommissioned. This small piece of open space cannot be reached from San Francisco (except by boat) without passing through Oakland and Alameda.
Michigan: The "Lost Peninsula" in Monroe County, Michigan, can only be reached via Toledo, Ohio. It is surrounded by Maumee Bay in Lake Erie. (Map)
Sandoval County, New Mexico: During World War II, Los Alamos County was created out of parts of Sandoval and Santa Fe Counties, for the convenience of the Manhattan Project.  That portion of Sandoval County which is within the San Ildefonso Indian Reservation, about 3 km², became an exclave bounded by Los Alamos County on the southwest, Santa Fe County on the east and Rio Arriba County on the north.

"Practical" exclaves
Some territories are not geographically detached from their motherland. Nevertheless, they are more easily reached by entering a foreign country. Very often, this is because of their location in a hilly area. It may also be because the only road enters that foreign place before coming back to the mother country.  These territories may be called "practical exclaves," "pene-exclaves" or "quasi-exclaves" and can be found along many borders, particularly those that are not heavily defended. They will only be attached to the motherland via an extremely small or thin slice of land.

Canada: St. Regis, Quebec: Part of the Akwesasne Mohawk Nation on the south shore of the Saint Lawrence River, it is attached by land to the St. Regis Mohawk Reservation in New York State; road access to the rest of Canada is only available through New York State.
United States:
 Alaska: The Alaska Panhandle, though connected geographically, is inaccessible by road from the rest of the state.  One must drive through Canada to reach the area from elsewhere in the state
Minnesota: The Northwest Angle is attached by land to the Canadian province of Manitoba; road access to the rest of the United States is only available through Manitoba
Washington: Point Roberts is at the southern end of a peninsula; land access to the rest of the United States is only available through the Canadian province of British Columbia

"Practical" subnational exclaves
United States:
Virginia :The Eastern Shore, comprising Accomack County and Northampton County, is located at the southern tip of the Delmarva Peninsula.  It is connected to the rest of Virginia via the 23 mile (37 km) long Chesapeake Bay Bridge-Tunnel.
Iowa: Carter Lake, a part of Pottawattamie County, Iowa, is surrounded by Omaha, Nebraska on three sides and the Missouri River on the fourth; road access to Iowa is only available through Nebraska.  
Michigan: The Upper Peninsula is attached by land to Wisconsin.  It can be reached from the Lower Peninsula of Michigan by the 5 mile (8 km) long Mackinac Bridge.

United Kingdom:
England: Pentreheyling can only be reached by road through Wales.

Historical exclaves
East Prussia, a German exclave during the Weimar Republic: it was separated from Germany after World War I, when Poland regained access to the Baltic sea.  East Prussia (essentially the old Duchy of Prussia) is now divided into Kaliningrad Oblast in Russia (see above) and the Warmian-Masurian Voivodship in Poland.
later became part of Poland and Russia (see Kaliningrad above).
East Bengal (1947-1955) / East Pakistan (1955-1971), now Bangladesh, was an exclave separated from West Pakistan by 1600 kilometres of India's territory. East Pakistan accounted for 70% of the exports of the country and was more populous than West Pakistan, and so in a sense it was actually the physically larger and politically more influential West that was the enclave and East Pakistan that was the Mainland.
Walvis Bay was a South African exclave in Namibia, before being incorporated with Namibia upon its independence.

Related pages
Enclave
Panhandle
List of countries that border only one other country

Other websites 
 Enclaves of the world 
 Jan S. Krogh's Geosite
 Article 'Tangled Territories' on European exclaves by Nicky Gardner of hidden europe magazine
 exclave.info