The Agatean Empire is a fictitous country that occupies the Counterweight Continent of Terry Pratchett's Discworld. It is the home of Twoflower and the Luggage.
It 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.
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), all jockeying to be emperor. They viewed politics as a bit like chess; the goal was to put your opponent in an untenable position, and the best way to do that was to take as many opposing peices as possible. And it didn't matter if you sacrificed your own pawns.
The Emperor himself 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.
(In The Colour of Magic and Mort the Emperor was a quite nice and idealistic young boy. He doesn't seem to have lasted long, although he outlived at least one Grand Vizier.)
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 reincarnation of One Sun Mirror (or rather, due to the complexities of Agatean belief, that One Sun Mirror would be the "preincarnation" of Cohen).
Since the events of The Last Hero, it can be presumed he's not the Emperor any more. Details as to his replacement are unknown.