Languages similar to the Aztec language have existed in Central Mexico for perhaps 1400 years. As early as 600AD, languages known as Nahuan were spoken by peoples in the area. It is believed that these language speakers came from the north in waves, settling in central Mexico.
Speakers of languages such as Nahuatl began to gain power, and by 1000AD (CE) it is likely that Nahuatl speakers were the dominant power.
One of the last Nahuatl speaking groups to come to the area was the Mexica, who would become a powerful force in the founding of the Aztec empire.
As the empire grew, so did the influence of Nahuatl (also called Classical Nahuatl, Mexicano or Aztec). Naturally, those who wanted to get along with the powers-that-were needed to speak it. It was a language of trade, and a language of prestige. It was used in literature extensively.
You can learn more about the letters of the Aztec language here. The Nahuatl language is an agglutinant language, which means that words and phrases are put together by combining prefixes, suffixes, and root words, in order to form an idea. For example, in Tetelcingo Nahuatl (a modern dialect), there is an 18-syllable word that means "you honourable people might have come along banging your noses so as to make them bleed, but in fact you didn't". That's right, you can just keep stringing those ideas together!
Actually, many forms of Nahuatl are still spoken today. It's likely that there were various dialects during the time of the Aztecs, just as there are today. Some dialects are so different that speakers can't understand one another!
You can hear what Nahuatl sounds like today. Here are some MP3s of a form of Nahuatl still spoken in the northern part of the State of Puebla in Mexico. Listen to Norte de Puebla Nahuatl.
Classical Nahuatl belongs to the Uto-Aztecan family of languages. These are found throughout the Western United States and Mexico today. Read more about the Aztec language (with a map) here. You can find extensive information about the whole Nahuatl family of languages here. You can even start to learn Nahuatl here!
For more, check out the references below.
References: An Analytical Dictionary of Nahuatl by Frances Karttunen; The Aztecs, 2nd Ed. by Michael E. Smith; The Aztecs by Richard F. Townsend; The Global Recordings Network; Introduction to Classical Nahuatl by J. Richard Andrews; Mexico: From the Olmecs to the Aztecs, 5th Ed. by Michael D. Coe and Rex Koontz; SIL article on the Nahuatl family; Nahuatl in the Middle Years: Language Contact Phenomena in Texts of the Colonial Period by Frances Karttunen