Top 10 Enclaves and Exclaves
An enclave is an entire country or part of a country completely surrounded by the territory of another country. The only way to access the territory of an enclave is by crossing the territory of the surrounding country because the enclave has no land connection to the rest of the main country territory.An exclave is "a piece of land which is politically attached to a larger piece but not physically coterminous (having the same borders) with it because of surrounding foreign territory." - wiki
Many entities are both enclaves and exclaves. But there are enclaves that are not exclaves and exclaves that are not enclaves. Examples:
Three sovereign countries are enclaves but not exclaves: Lesotho, San Marino and Vatican City.
Exclaves which are not enclaves: Nakhichivan Autonomous Republic, and some smaller entities.

The Kingdom of Lesotho is world's largest enclave that is entirely surrounded by South Africa. Lesotho gained its independence from the United Kingdom in 1966 (South Africa was already independent).
The kingdom in the Sky...

It's as beautiful as Italy. The nation has such a clean look. The nation may seem small but when you cross it you can see that it isn't that tiny. It took me 1:30 hours to cross the nation. So glad I went there.
The republic of San Marino is an enclave within Italy.

An enclave within the city of Rome, Italy, established in 1929.

Baarle-Hertog is a Flemish municipality of Belgium, much of which consists of a number of small Belgian exclaves in the Netherlands.
Read this and try to imagine how these bits of land look like - you can cross 4 international borders in just 4 minutes:
"Baarle-Hertog is noted for its complicated borders with Baarle-Nassau, Netherlands. In total it consists of 24 separate parcels of land. The main division of Baarle-Hertog is Zondereigen (named after its main hamlet), located north of the Belgian town of Merksplas. In addition there are twenty Belgian exclaves in the Netherlands and three other sections on the Dutch-Belgian border. There are also seven Dutch exclaves within the Belgian exclaves (I.e., counter-exclaves). Six of these Dutch enclaves are located within the largest Belgian enclave, and a seventh in the second-largest Belgian enclave. An eighth Dutch exclave is located nearby Ginhoven." - wiki
"Some houses in the town of Baarle-Hertog/Baarle-Nassau are divided ...more
In the image you see the border - if you have dinner on the Dutch side (Nl, left) and your restaurant closes earlier, you just have to move to the Belgian side (B, right), which is a step away, and finish your dinner.
Luckly, both of these countries are in the European Union so you can freely travel between them, both also use Euros and both speak the same language (Dutch)

Naxcivan aka Nakhchivan is a pretty big chunk of Azerbaijan in Armenia. To access this exclave from Azerbaijan you have to cross either Armenia or Iran (both routes are too close to the dangerous region of Nagorno-Karabakh).

It's a very beautiful Italian municipality entirely surrounded by Switzerland (that's a "true" enclave and exclave of Italy).
How did this happen? In 1798, the canton of Ticino chose to become part of the Swiss Confederation, but this municipality chose to remain part of Lombardy, Italy. This decision was confirmed by a referendum in 1814. Later Campione d'Italia tried to join Switzerland but the Swiss refused, not wanting to be drawn into Italian disputes.
So how Italian is Campione d'Italia today? It has a Swiss dialing code, Swiss emergency responders, and uses the Swiss currency - Swiss Franc.
It's basically italy but you don't have to pay tax (not like italians pay taxes anyway)

West Berlin was a West German territory located in the middle of East Germany

Sokh aka So'x is a large enclave of Uzbekistan in Kyrgyzstan. But it's with 99% Tajik population and only 1% Kyrgyz, and almost no ethnic Uzbeks.
Hatta is the inland exclave of the emirate of Dubai in the United Arab Emirates.

A mapping error during the 1783 Treaty of Paris left a chunk of US territory in present day Canada, then British territory. The map they used misrepresented the lake's location and the border drawn between the US and what was then British territory, created this oddity.

Ceuta is a Spanish autonomous city in Morocco, Northern Africa!
Spain erected 6-meter high fences topped with barbed wire to keep out illegal immigrants from Morocco. At night these barbed wire fences are lit up and create a beautiful spectacle.
It's a German town entirely surrounded by Switzerland. Population - about 1,450 inhabitants