Suzerainty
Control of a weaker state's foreign relations while leaving it internally self-governing, the standard imperial arrangement before formal sovereignty.
Suzerainty is the arrangement in which a powerful state controls a weaker one's external affairs, above all its right to make war and treaties, while leaving its ruler on the throne and its internal government alone. The weaker party is a vassal, not a province. It has a flag, a king, its own laws and taxes, and no independent foreign policy. For most of recorded history this was the normal way large powers organized their peripheries, and full sovereignty for everyone was the exception.
Why it was preferred
Direct rule is expensive. It requires administrators who speak the language, garrisons in every district, and responsibility for famine, justice, and roads. Suzerainty outsources all of that to a local ruler who already has legitimacy, and buys only the thing the imperial power actually wants: a guarantee that this territory will not host a rival, and usually tribute.
The Chinese tributary system is the most fully theorized version. Korea, Vietnam, Siam, and others sent periodic missions to the emperor with gifts, performed the ritual prostration, received investiture and more valuable gifts in return, and governed themselves entirely. Beijing did not tax them or station troops. It received acknowledgment of hierarchy, which in the Confucian conception of world order was the substance rather than the decoration.
Gillray's famous caricature of Lord Macartney's 1792 embassy captures the collision that followed. The British came to Beijing as one sovereign's representatives to another and wanted resident ambassadors and open ports. The Qing court could only process them as tribute bearers. Neither side was confused about the goods on offer; they were arguing about what kind of relationship existed at all.
The imperial workhorse
European empires used it constantly. Roughly forty percent of the Indian subcontinent, more than five hundred princely states, was never directly ruled by the British at all. Their maharajas and nizams kept their thrones under British paramountcy, with a Resident at court, no right to deal with foreign powers, and no army worth the name. It was cheaper than administering the territory, and it created a class of rulers whose position depended on the imperial power surviving.
The Ottomans ran Egypt, Tunisia, and the Danubian principalities this way for long stretches. The Soviet Union ran the Warsaw Pact this way after 1945: sovereign states with their own flags, seats at the United Nations, and no ability to leave the alliance, as Hungary discovered in 1956 and Czechoslovakia in 1968.
Why the word is still fought over
Modern international law does not really have a category for it. States are sovereign or they are not, which makes suzerainty a term you now find mostly in arguments about status.
Britain spent the twentieth century describing China's relationship to Tibet as suzerainty rather than sovereignty, a formula that conceded Chinese primacy while withholding ownership, and quietly abandoned the distinction in 2008. Bhutan's 1949 treaty with India committed it to be guided by India in foreign relations, wording revised in 2007. The debates over Taiwan, Kosovo, and Abkhazia are arguments about the same gap between who governs a place and who is entitled to speak for it.
The concept is worth keeping because it names something real that the sovereign-or-not binary hides. A great deal of contemporary influence, basing agreements, security guarantees, debt leverage, and defense dependencies, produces states that are formally sovereign and practically unable to choose their alignment. That is suzerainty with better manners.