From Wikipedia

Aztecs
Inside The Hidden Kingdom

In the 13th century in the Valley of Mexico, there existed many city-states including Chalco, Xochimilco, Tlacopan, Culhuacan, and Atzcapotzalco. The most powerful were Culhuacan on the south shore of Lake Texcoco and Azcapotzalco on the west shore.

As a result, when the Mexica arrived in the Valley of Mexico as a semi-nomadic tribe, they had nowhere to go. In roughly 1248, they first settled on Chapultepec, a hill on the west shore of Lake Texcoco, the site of numerous springs.

In time, the Tepanecs of Azcapotzalco ousted the Mexica from Chapultepec and the ruler of Culhuacan, Cocoxtli, gave the Mexica permission to settle in the empty barrens of Tizaapan in 1299. There they married and assimilated into Culhuacan culture.

In 1323, they asked the new ruler of Culhuacan, Achicometl, for his daughter, in order to make her the goddess Yaocihuatl. Unbeknownst to the king, the Mexica actually planned to sacrifice her. The Mexica believed that by doing this the princess would join the gods as a deity. As the story goes, during a festival dinner, a priest came out wearing her flayed skin as part of the ritual. Upon seeing this, the king and the people of Culhuacan were horrified and expelled the Mexica.

Forced to flee, in 1325 they went to a small island on the west side of Lake Texcoco, where they began to build their city Tenochtitlan, eventually creating a large artificial island. It is said that the Aztec god, Huitzlipochtli, instructed the Aztecs to found their city at the location where they saw an eagle, on a cactus, with a snake in its talons. The Aztecs, apparently, saw this vision on the small island where Tenochtitlan was founded.

Another Mexica group settled on the north side of this island: this would become the city of Tlatelolco. Originally, this was an independent Mexica kingdom, but eventually it was absorbed by Tenochtitlan, and treated as a "fifth" quadrant. The famous marketplace described by Hernan Cortés and Bernal Diaz del Castillo was actually located in Tlatelolco.

In 1376, the Mexica elected their first tlatoani, Acamapichtli, following customs learned from the Culhuacan.