Metropolis, a Greek word actually consisting of two words - meter (mother) and polis (city). Hence, metropolis can be translated into English as mother city. Curiously, some languages prefer an even stronger translation: In Czech, for example, they like to refer to their own metropolis as Praha, matka mest, which literally means, Prague, mother of cities.
In the ancient times, the term metropolis was reserved to a specific type of a city: The capital of an empire, state, kingdom and other places from which "the rest of the world" (or at least some major territory) was ruled. Thus a metropolis would rank considerably higher than a provincial city.
In modern days, any major city likes to be thought of as a metropolis, even if it is not the seat of the government. This is true particularly of the cities in the United States because a typical US State Capital is not a big city. By the same token, most US States have at least one big city which is not the Capital of that State.
Furthermore, at least in the US, the population in the area surrounding a major city is typically much larger than the population of the city itself. The two populations are typically added up and published as the population of the greater metropolitan area.
Naturally, if we follow the ancient tradition, any State Capital still deserves the title of metropolis even if it is not the cultural and business center of that state.
So, a modern meaning of the word metropolis often is any place which is a political, cultural, or economic center, though not necessarily all of the above (we might even think of E2 as a virtual metropolis).
Curiously, there is a Metropolis association which accepts any city as its member if it is a capital of a country or if it has at least one million inhabitants. It calls itself the World Association of the major metropolises. It is a somewhat strange association in that no US city is a member, nor are any of the four capitals of Central European countries (Prague, Vienna, Bratislava, Budapest.
Editorial: In my opinion calling oneself the World Association of the major metropolises is insensitive at least, even outright offensive to non-members. For one thing, a city either is a metropolis, or it is not, there are no major ones, and certainly not the major ones. And if there were any such things, certainly at least some of the cities in the US and the capitals of Central European countries would qualify. You can see a world map with the member cities listed at metropolis.org.