1. Anáhuac: This was the name given by the Nahua people to the central region of Mexico, which was the heart of the Aztec Empire. The word "Anáhuac" is derived from the Nahuatl word "āno-huac", meaning "near the water."
2. Aztlan: Aztlan was the legendary homeland of the Aztecs, from where they migrated to central Mexico in the 13th century. The word "Aztlan" is derived from the Nahuatl word "aztatl", meaning "heron."
3. Tenochtitlan: Tenochtitlan was the capital city of the Aztec Empire, located on the site of modern-day Mexico City. The city was founded in 1325 by the Aztecs and became one of the largest and most powerful cities in the pre-Columbian Americas.
4. New Spain: After the Spanish conquest of the Aztec Empire in the 16th century, the region became known as New Spain (Nueva España in Spanish). This name was used for the Spanish colony that encompassed much of what is now Mexico, as well as parts of the United States and Central America.
5. United Mexican States: The name "United Mexican States" was adopted in 1824 when Mexico gained its independence from Spain. The name reflects the country's federal system of government and the union of different states that make up the nation.
These names represent different periods in Mexican history and reflect the diverse cultures and civilizations that have shaped the country's heritage and identity.