A lot of countries have embassies. A lot have an embassy in DC.

And?

Not much. Mongolia is a country, so will have embassies in major countries. I've walked past the one in London.

An embassy is a building which serves as a country's headquarters in that country. It's where they do official business from and so on.

It means they are part of the group of countries that do:

As was mentioned, an embassy is the headquarters of a foreign country. As a general rule, embassies are located in capital cities i.e. Washington DC, London, Paris, Moscow, etc. What you see in other cities like Miami, LA, Chicago, etc. are known as consulates. These buildings are simply local branches of the embassy. For example, there is a Belgian consulate here in Tampa with the embassy in DC.

It means that the building and land where the building is, is technically Mongolia.

All countries swap a little bit of their own country for another so that they have posts round the world. Diplomats are exchanged too and they all have special status. Diplomats are actually above the law in the land they are living and only subject to the law of their own land.

It means that a representative of every country is already installed in every other country.

This was an established method long before we had the transport and communication facilities we now have.

Each Embassy is autonomous and the police can not violate the building. In some incidents over the years criminals have taken sanctuary in foreign embassies. Protected from prosecution, they could be flown home if they had this 'Diplomatic Immunity' status.

It is quite an interesting subject. Worth looking up.

An Embassy is the official representation of one country's government to another's.

As such it is normally headed by an Ambassador, and is almost always in the capital city of the other country. The term embassy is also used to describe the grounds or facility that the Embassy uses as its offices and sometimes living quarters.

Embassies are for government to government relationships; consulates are for individuals in another country to do business with a foreign government. For example, Chinese people in the US get services from the Chinese consulates, and anyone in the US wishing to get a visa to visit China would contact the Chines consulates as well.