The Agatean Empire is a fictitious country that occupies the equally fictitious Counterweight Continent of Terry Pratchett's Discworld. It is the home of Twoflower and the Luggage. The country was first mentioned in the very first Discworld novel The Colour of Magic, but was not explored: its main role was as the place from which Twoflower had come. It appeared in a brief segment of Mort, and was the setting for most of Interesting Times.
The Agatean Empire is similar to feudal Japan or Imperial China, most obviously in the vast wall that surrounds it, and the belief that anyone from outside the Empire must be an invisible vampire ghost (probably a play on gwailo, lit. ghost man). The name is a pun on the term "Jade Empire," referring to ancient China.
The crust of the Agatean Empire is composed largely of gold, octiron and other heavy metals. In Ankh-Morpork and its environs, the Empire is known as the Aurient - 'the place where the gold comes from' (a pun on Orient and aurum, the Latin for gold). Exchange rates between the Ankh-Morpork dollar and the Empire's currency, the rhinu, has never been established because the solid gold coins are worth significantly less on the Counterweight Continent than on the Sto Plains.
The Empire was founded by the Emperor One Sun Mirror thousands of years ago and until fairly recently it was ruled by a number of constantly fighting and somewhat psychotic noble families (the Hongs, the Sungs, the Fangs, the Tangs and the McSweeneys (very old and established family)), all jockeying to be emperor. They viewed politics as similar to chess; the goal was to put one's opponent in an untenable position, and the best way to do that was to take as many opposing pieces as possible. And it didn't matter if you sacrificed your own pawns.
As in our own feudal Japan, the Emperor was seen as a god, and could do anything he liked. Given how nastily inventive a noble had to be to get to this point, this was not a good thing.
During Interesting Times, Cohen the Barbarian was declared Emperor, and started changing the system into one a no-nonsense barbarian could feel comfortable with. It was felt by many of the peasantry that he was the "preincarnation" of One Sun Mirror (because Agateans believe in a form of backward reincarnation in which the soul's next life takes place chronologically earlier than its previous life). Since the events of The Last Hero, it can be presumed Cohen is no longer Emperor. Details of his replacement are unknown, although it is implied that most of the duties of governance had already been assumed by his Grand Vizier Twoflower by the time of his intended-to-be death.