In Tolkien-influenced high fantasy, realm is generally used in place of country, and means the same thing.
A kingdom is a country which is specifically ruled by a monarchy. An empire can be one country or a collection of countries and territories, ruled by an emperor/empress.
You can have one king or queen (or prince/princess) who rules a group of kingdoms; this will be reasonably referred to as The Five Kingdoms (if there's only one such related set) or The Three Kingdoms of Gazornenplatz (as opposed to The Three Kingdoms of Katzenjammer).
Cities, towns, and villages are all collections of people living and working, and the main difference is the size. They are independent of one another; you don't have a town inside a city. You can have neighborhoods or boroughs, which are smaller divisions of cities (Park Slope is a neighborhood in Brooklyn; Brooklyn is a borough of New York City), but those are societal and geographical sections of one larger political municipality.
You've missed state and province in your list, which are both medium-sized political divisions of countries. (They can technically be any percentage of a country.)
You can also have territory, principality, and colony, which can be anywhere from village-sized to country-sized, and can be attached to or independent of any other political entity. A city with a big enough army and enough resources can be a city-state (which is literally a country which is just the bounds of one city; think of the Vatican or Monaco).