The Aztec Empire, or the Triple Alliance (Classical Nahuatl: Ēxcān Tlahtōlōyān, [ˈjéːʃkaːn̥ t͡ɬaʔtoːˈlóːjaːn̥]), began as an alliance of three Nahua altepetl city-states: Tenochtitlan, Texcoco, and Tlacopan. These three city-states ruled the area in and around the Valley of Mexico from 1428 until the combined forces of the Spanish conquistadores and their native allies under Hernán Cortés defeated them in 1521.
The Triple Alliance was formed from the victorious factions in a civil war fought between the city of Azcapotzalco and its former tributary provinces. Despite the initial conception of the empire as an alliance of three self-governed city-states, Tenochtitlan quickly became dominant militarily. By the time the Spanish arrived in 1519, the lands of the Alliance were effectively ruled from Tenochtitlan, while the other partners in the alliance had taken subsidiary roles. The alliance waged wars of conquest and expanded rapidly after its formation. At its height, the alliance controlled most of central Mexico as well as some more distant territories within Mesoamerica. The Aztecs left rulers of conquered cities in power so long as they agreed to pay semi-annual tribute to the Alliance, as well as supply military forces when needed for the Aztec war efforts.
Aztec Empire in "Eyewear"
In the world known to the Moorish slave Estevánico, the Aztec had been conquered by Spain over a decade prior to his discovery of his fantastic eyewear. Esperanza/Amal confirmed this, and that at least part of the conquest's success was do to the fact that Hernán Cortés had a device similar to Estevánico's eyewear. Esperanza/Amal surprised Estevánico further by telling him that Montezuma himself had given the device to Cortés.
Esperanza/Amal also warned Estevánico that the Aztec had learned much in 700 years. Estevánico did not fully grasp what this warning meant until, while residing in Mexico City, he was pursued by Aztecs who spoke a language that sounded faintly like Nahuatl.
What Estevánico never quite knew, and what Esperanza could not tell him, was that while in the Aztec did fall in multiple timelines, in at least one, they defeated the Spanish and built a formidable empire. In the 23rd century of this timeline, the Aztec were locked in a battle with their rival, the Inca, also a formidable empire, and that their fight crossed back across both timelines and time itself.
Aztec Empire in "The Man who Came Late"
When Sir Holger told Alianora and her family of his adventures in the many worlds and timelines he had visited, he mentioned a near-fatal encounter with the "feathered demons, or pagan gods" of the Aztecs, who ate hearts and drunk blood. This story made Alianora shiver.
An earlier reference in Poul Anderson's A Midsummer Tempest confirms that these were indeed Aztec gods.
Aztec Empire in "The Pugnacious Peacemaker"
Aztecia was one of the last nations in the world to abolish the practice of human sacrifice.
While Aztecia is also mentioned in L. Sprague de Camp's "The Wheels of If," the bit about human sacrifice is Harry Turtledove's embellishment. Aztecia plays no role beyond a passing reference in either story.
Aztec Empire in Through Darkest Europe
Nahuatl was a common language in a certain region of the Sunset Lands.