It's clear from the cities with tens of thousands of people, especially around lake Texcoco, that farming was a large portion of life. How was it done?
How similar was it to European agriculture? How much more difficult was it, without (as far as I know) domesticated draft animals?
Around lake Texcoco the main method for large-scale agriculture was the Chinampa, which used small, rectangular areas of fertile arable land to grow crops on the shallow lake beds. They literallywere small artificial islands fenced with wattel and layered with mud, lake sediment, and decaying vegetation. The crop yeild of the chinampa was relatively high, having around 7 crops a year. Also agriculturue was a whole community effort, the land was owned by the calpulli (neighborhood) and each family was assigned a part of land to be cultivated, with the yeilds going first to be used by the assigned family and then all that was left (which was a lot) went to the calpulli, to be traded for other goods or stored.
Here some links to further reading :)
http://www.sscnet.ucla.edu/ioa/backdirt/Fallwinter00/farming.html
http://northeasternpermaculture.wikispaces.com/file/view/mx_city_floating_gardens.pdf
Also this book is really good for basic understanding of Mesoamerican day to day life, but I think it is only in spanish :S
Mesoamerican agriculture centered around three staple crops: maize, beans, and squash. Usually, these were all planted in the same agricultural plots. Beans could be planted alongside maize and the vines would essentially climb up the maize stalk. Since maize is nitrogen leaching, and beans are nitrogen fixing, the two plants complemented each other and if planted together reduced the nutritional strain on the soil. Additionally, if maize is soaked in lime and eaten within a few hours of beans, it can provide all eight essential amino acids. Supplementary crops included numerous kinds of tropical fruit, avocados, and peppers. Because Mesoamericans didn't have draft animals or plows, the tool of choice was a digging stick. This spear-like tool was used to pierce the ground making a hole at regular intervals, into which seeds were placed.
Domestic meat sources were typically limited to dogs (which were eaten) and turkeys, although in some cases people had ducks. Meat could also be supplemented with wild game and fish. Fish was particularly important to cultures along the coast (like the Olmec or the Mokaya) as well as those centered around highland lakes (especially the Tarascans).
In the Lowlands
There were a few different strategies to farming, as others have pointed out in the thread. Milpa agriculture was common in the tropical lowlands (such as the Olmec or Maya). This is essentially a form of Slash-and-Burn agriculture. You'd cut down all of the plants in a particular plot of land, although trees would be left standing for shade (and in some cases fruit trees were planted for this purpose).
The biomass from the slashed plants was then burned to produce a layer of fertile ash for growing crops. Rainforest soils are extremely thin; you typically have about 10cm of fertile humus on top of nutritionally sterile sediment. Most of the nutrients in rainforests is caught up in the plant matter, so burning this released the nutrients and made them available for farming. This worked well, but you only got about 2 seasons of farming from a burned plot before the land had to be abandoned. The old plot would be left fallow (wild) and the tropical would essentially reclaim it while they farmed a different plot. After about 7-8 years or so, they would return to the old plot and begin again.
In the Mountains
Along lakeshores in the Central Mexican Plateau, people dug a series of canals along the edges of the lake and piled up the dug-out sediment in the plots between canals. The raised area was called a "chinampa," and these plots were often stabilized by planting fruit-bearing trees in the corners. The trinity crops (maize, beans, squash) were grown in the middle, and water was readily available from the canals. Fish would often use the canals as well, which meant farmers had access to maize, beans, squash, fruit, and fish all within the same plot. Canoes would be used to move between plots via the canals, and this provided a way to transport agricultural goods back and forth from the plots to markets.
The highlands were also marked by extensive use of agricultural terraces. Hillside terraces were constructed by building a series of walls along hill slopes creating a series of flat platforms on which crops could be grown. The standard trinity crops were grown on hillside terraces, but this technique was also the farming method of choice for growing maguey, which was used to make fabric as well as an alcoholic beverage called pulque.
There was also a technique called lama-boro terracing, which involved constructing a series of miniature dams along a stream or river. These dams trapped sediment flowing downstream and transformed sloped, v-shaped river beds into a series of flat platforms functionally similar to hillside terraces. In highland areas of Oaxaca, in particular, a combination of hillside terraces and lama-boro terraces transformed the extremely rugged mountains into a highly manufactured landscape suitable for agriculture.
In most Mesoamerican cultures, a combination of the strategies described above was used, often with local variations.
Sources:
Everything described above comes from books in the appropriate section of the AskHistorians book list.
Specifically, information on Maya agriculture can be found in Demarest 2004, Aztec agriculture from Smith 1996, and other agricultural practices are described by Evans 2008.
Not a historian, but as a Mexican I thought I could chime in a bit since no one has answered. The Aztecs in Tenochtitlan used a system called chinampas. Basically they would make a type of artificial islands in the lake of Texcoco in which they would rise the level of the land above water to plant. I don't know the yield, but the newly created areas were very fertile. A difference from Europe, which still persists today is planting corn instead of wheat. Beans were and still are a popular crop.
Outside of the Central Valley, especially in the more heavily forested south and southeastern regions, Milpa agriculture was a common form of heavy agriculture, first among the Olmec then among the Maya. This was a form of slash and burn agriculture where the farmer would clear a field from the edge of the jungle, grow on it for two years, then leave it fallow for 7 or 8 years. At lower population levels, this is a sustainable form of agriculture, but as population density increase, it becomes less and less sustainable. Here is a link to a decent Wiki on the topic (http://en.wikipedia.org/wiki/Milpa)
Qualifications: MA Anthropology focused on Mesoamerican Middle Formative period art and culture