Beginning in the 1100s AD, probably because the Medieval Warming Period made it too hard to live where they were, the Mexica people had moved down to what is now Mexico from further north in North America. The Mexica may have been related to the Shoshone and the Ute people who lived among the Rocky Mountains. After some moving around Mexico in the 1200s, by 1325 the Mexica settled down with their capital city at Tenochtitlan. Around this time, the Maya kingdom to their south began to lose power, and the small kingdoms of Mexico began to get stronger. But the Mexica were still less powerful than their neighbors, the Tepanecs. Like other newcomers - the Greeks in Egypt, the Ostrogoths in Europe, the Arabs and then the Turks in West Asia - the Mexica hired themselves out as mercenaries to stronger states like the Tepanecs.
The Mexica had kings for about 150 years, and they were all from the same family. The first king of the Mexica was Acamapichitli, who was elected in 1376. His son, Huitziláihuitl, succeeded him in 1395, and then his grandson (we think), Chimalpopoca, in 1417. Chimalpopoca and his people fought as mercenaries for the Tepanecs, but then in 1427 Chimalpopoca was killed apparently by the Tepanec king Maxtla, who was trying to get more control over the Mexica.
After Maxtla killed Chimalpopocaa, a more distant relative became king, Itzcóatl. Itzcóatl, perhaps not surprisingly, didn't want to have anything more to do with the Tepanecs, and the Mexica became more independent. They thought of themselves as making a fresh start. Under Itzcóatl, the Mexica formed the Triple Alliance with their neighbors, and the whole group together was known as the Aztecs. The Aztecs began to be more powerful than their old neighbors (this is also the usual thing that happens with newcomers who fight as mercenaries). Itzcóatl's nephew, Montezuma, succeeded him in 1449. Montezuma was a great ruler, and under his rule the Aztecs conquered a lot more land. To show his power, Montezuma built big zoos and botanical gardens in his palace, where he could show off all the different animals and plants that he ruled over. But also during Moctezuma's reign, there was a terrible drought and people went hungry. Montezuma began to sacrifice a lot of people to the gods - prisoners of war taken in the "Flower Wars," which were a sort of pretend wars just in order to get prisoners of war to sacrifice.
Montezuma ruled a long time and died old. His grandson, Axayacatl, became king after him. There were rebellions, and Axayacatl had to reconquer some Mexica land. When Azayacatl died in 1481, his brother Tizoc became king, though apparently he didn't accomplish much. He seems to have been killed by his own people after five years. Finally their younger brother Ahuizotl became king in 1486. Under Ahuizotl, the Mexica conquered even more land, further south all the way to Maya territory (modern Guatemala). Ahuizotl's son Montezuma II succeeded him in 1502.