Short answers, pulled from the story.
A city-state is an independent sovereign city that serves as the primary hub of political, economic, and cultural life within its contiguous territory. It contrasts with a regular country, which usually combines a capital city with additional urban centers and countryside.
Singapore, Monaco, and Vatican City are the candidates most commonly discussed as modern sovereign city-states. Singapore is the largest and most populous, holding full sovereignty, its own currency, a robust military, and substantial international influence.
Vatican City was recognized as an independent state in 1929 through the Lateran Treaties negotiated by Benito Mussolini between King Victor Emmanuel III and Pope Pius XI. With fewer than 1,000 residents inside a 0.49 square kilometre enclave, it is by far the smallest sovereign country in the world.
The ancient Greek poleis such as Athens, Sparta, Thebes, and Corinth, and the merchant city-states of Renaissance Italy such as Florence, Venice, Genoa, and Milan, are among the best known. The Roman Republic also began as a city-state before growing into a vast empire.
Singapore was part of the Federation of Malaysia for two years before it seceded in 1965, becoming an independent republic, a city, and a sovereign country at once. Six million people live within 728.3 square kilometres, making it the 2nd-most-densely populated country in the world after Monaco.
Hong Kong and Macau function as city-states as special administrative regions of China, while Gibraltar is a British Overseas Territory described in similar terms. Cities that are constituent states of a federation, such as Bremen, Berlin, Hamburg, Vienna, and Brussels, also qualify as non-sovereign city-states.
The Free City of Danzig existed from 1920 to 1939, the Free State of Fiume from 1920 to 1924, and the Shanghai International Settlement from 1845 to 1943. The Free Territory of Trieste sat under the UN Security Council from 1947 to 1954, and West Berlin was ruled by the Western Allies from 1948 to 1990.