Short answers, pulled from the story.
The Aztecs were a Mesoamerican civilization that flourished in central Mexico from 1300 to 1521. The term refers to several Nahuatl-speaking peoples of central Mexico, especially the Mexica of Tenochtitlan, and the broader Nahua polities sharing their cultural traits.
The Aztec Empire originated in 1427 as the Triple Alliance of the city-states Tenochtitlan, Texcoco, and Tlacopan. In 1428 the alliance besieged and destroyed Azcapotzalco, making Tenochtitlan the dominant power in the Valley of Mexico.
The Mexica founded Tenochtitlan on a small swampy island in Lake Texcoco, usually dated to 1325. According to legend, in 1323 they saw a vision of an eagle perched on a prickly pear cactus eating a snake, which indicated where to build.
Alexander von Humboldt originated the modern usage of Aztec in 1810 as a collective term for peoples linked to the Mexica state and the Triple Alliance. After William H. Prescott published his work on the conquest of Mexico in 1843, the term was adopted across most of the world.
Hernan Cortes allied with enemies of the Mexica, particularly the Tlaxcalteca, and entered Tenochtitlan in 1519. After the capital fell and the emperor Cuauhtemoc was captured on the 13th of August 1521, the Spanish founded Mexico City on the ruins of Tenochtitlan.
The Aztecs believed humans were responsible for the sun's continued revival and for paying the earth for its fertility, so sacrifice was seen as sustaining creation. Many scholars also point to ideological reasons, since the public sacrifice of captured warriors displayed political power and deterred rebellion by subjugated polities.