The Olmec
The Olmec civilization's roots go back to around 1500 B.C. and they continued to flourish until roughly 400 B.C. The Olmec settled in swampy, lowland rivers located near the Gulf of Mexico. Not much was known about the Olmec until a pair of excavations in San Lorenzo and La Venta in the 1930's. These excavations revealed everything that we know about the Olmec today.
The Olmec carved numerous large stone heads; some as big as 9 feet tall and weighed up to 40 tons. The heads are believed to be remnants of large statues devoted to the Olmec rulers. There is record of a lot of trade between the Olmec and various other early Mesoamerican tribes. One of the only other details known about the Olmec is that they practiced a procedure called "Slash and Burn Farming" which entailed burning down large areas of a forest at a time, and farming in the fertile ashes left behind.
The Mayans
The Mayans thrived from 900 B.C. to 900 A.D. in the Yucatan Peninsula (present day Mexico). The Mayans, unlike many other civilizations at the time, were not one unified country, but rather, several waring city-states )similar to Ancient Greece). Due to the varied environment of Mexico, the Mayans lived in highlands, lowlands, and coastal plains, and were very successful in adapting to their environment.
The Mayans were a very intelligent civilization and had many useful ideas and inventions. For example, they created the first 365 day calendar, which predicts that the world will end on Dec. 21, 2012. They also created one of the world's first written languages, as well as an advanced number system. Their priests made various astronomical discoveries, which they believed to hold strong religious significance. The Mayans also used some of the world's first bound books.
Agriculture and agricultural trade were at the forefront of the Mayan economy. The Mayans traded agricultural products for basically whatever they needed in return. Some of this trading was done overseas, and the traders used canoes for transportation.
The Aztec
The Aztec civilization began in 1200 A.D. in central Mexico (where Mexico City is located today). Their capital was called Tenochtitlan, and it was the agricultural center of the entire empire.
Because of the lack of fertile land in and around Tenochtitlan, the Aztec were forced to find a way to grow crops on the land that they did have. The solution that they came up with was the invention of the Chinampa, an artificial agricultural island. Chinampas were created by compacting mud together and floated on top of a small body of water, such as a pond or a flooded part of a city. The farmers planted crops in the soil on a Chinampa and allowed them to grow.
The Aztec were also the first Mesoamericans to vastly exend their border, all the way south to the border of Guatemala. The newly conquered land allowed for the population of Tenochtitlan to grow, exponentially, to over 200,000 people.
The Incas
The Incas lived in Peru from 100 A.D. to 1531 A.D. They named their capital Cuzco. The first well known Incan emperor, Pachacuti, wasn't in power until late in the civilization's history. In 1438, Pachacuti conquered many smaller tribes and was able to extend the Incan border vastly to the North and South. This conquest was even compared to those of Alexander the Great. After his conquests, Pachacuti established a very strong central government, where he and his advisors held the majority of the power. He also brought with the establishment of the first common Incan language, Quechua.
Instead of writing things down, the Incans developed a series of different lengths and colors of rope to represent words and numbers.
Finally, the Incan civilization was defeated by spanish conquistadors in 1531, and were forced to surrender all of their land, and relinquish government control to Spain.