Answer by taking a quote from the following article:

The Aztecs (), were a Mesoamerican culture that flourished in central Mexico in the post-classic period from 1300 to 1521, during the time in which a triple alliance of the Mexica, Texcoca and Tepaneca tribes established the Aztec empire. The Aztec people were certain ethnic groups of central Mexico, particularly those groups who spoke the Nahuatl language and who dominated large parts of Mesoamerica from the 14th to the 16th centuries. Aztec culture is the culture of the people referred to as Aztecs, but since most ethnic groups of central Mexico in the postclassic period shared basic cultural traits, many of the traits that characterize Aztec culture cannot be said to be exclusive to the Aztecs. For the same reason, the notion of "Aztec civilization" is best understood as a particular horizon of a general Mesoamerican civilization.

The Nahuatl words aztecatl [as'tekatl] (singular) and aztecah [as'teka?] (plural) mean "people from Aztlan", a mythological place for the Nahuatl-speaking culture of the time, and later adopted as the word to define the Mexica people. Often the term "Aztec" refers exclusively to the Mexica people of Tenochtitlan (now the location of Mexico City), situated on an island in Lake Texcoco, who referred to themselves as Mexihcah Tenochcah [me:'Si?ka? te'notSka?] or Colhuah Mexihcah ['ko:lwa? me:'Si?ka?].  Sometimes the term also includes the inhabitants of Tenochtitlan's two principal allied city-states, the Acolhuas of Texcoco and the Tepanecs of Tlacopan, who together with the Mexica formed the Aztec Triple Alliance that controlled what is often known as the "Aztec Empire". In other contexts, Aztec may refer to all the various city states and their peoples, who shared large parts of their ethnic history and cultural traits with the Mexica, Acolhua and Tepanecs, and who often also used the Nahuatl language as a lingua franca. In this meaning, it is possible to talk about an Aztec civilization including all the particular cultural patterns common for most of the peoples inhabiting central Mexico in the late postclassic period.  When used to describe ethnic groups, the term "Aztec" refers to several Nahuatl-speaking peoples of central Mexico in the postclassic period of Mesoamerican chronology, especially the Mexica, the ethnic group that had a leading role in establishing the hegemonic empire based at Tenochtitlan. The term extends to further ethnic groups associated with the Aztec empire, such as the Acolhua, the Tepanec and others that were incorporated into the empire. In older usage the term was commonly used about modern Nahuatl-speaking ethnic groups, as Nahuatl was previously referred to as the "Aztec language". In recent usage, these ethnic groups are referred to as the Nahua peoples. Linguistically, the term "Aztecan" is still used about the branch of the Uto-Aztecan languages (also sometimes called the yuto-nahuan languages) that includes the Nahuatl language and its closest relatives Pochutec and Pipil.  To the Aztecs themselves the word "aztec" was not an endonym for any particular ethnic group. Rather, it was an umbrella term used to refer to several ethnic groups, not all of them Nahuatl-speaking, that claimed heritage from the mythic place of origin, Aztlan. In the Nahuatl language "aztecatl" means "person from Aztlan". Alexander von Humboldt originated the modern usage of "Aztec" in 1810, as a collective term applied to all the people linked by trade, custom, religion, and language to the Mexica state and the Triple Alliance. In 1843, with the publication of the work of William H. Prescott, the term was adopted by most of the world, including 19th-century Mexican scholars who saw it as a way to distinguish present-day Mexicans from pre-conquest Mexicans. This usage has been the subject of debate in more recent years, but the term "Aztec" is still more common.

What pricipals called
Aztec Triple Alliance