Chocolate and Tobacco. Both of these were integral to shamanic rituals, and their culture generally.

Tobacco was considered to be the physical incarnation of the God Cihuacoahuatl and was consumed in almost every possible way imaginable, from smoking, to snuffing, to drinking, and even rectally. Some Aztec shaman were reported to pass out/ die from nicotine poisoning from rectally ingested (right word?!) tobacco. Tobacco was used as an appetite suppressant on hunting trips to decrease the need to stop and eat.

Similarly, Chocolate was integral to their culture. Very different from the sweetened solid we eat today, it was primarily consumed as a cold beverage mixed with spices including chilli power. The cocoa bean was Aztec currency and Montezuma possessed over 9 billion beans upon Cortez's arrival. The drinking of chocolate was a symbol of status and wealth, and the Aztecs implemented vast trade routes from the Amazon to present day Mexico from which to procure their beans.

Needless to say these two plants are integral to modern society as they are the two most commonly consumed drugs on the planet.

Source: Marcy Norton, Sacred Gifts, Profane Pleasures: A History of Tobacco and Chocolate in the Atlantic World

EDIT: With regards to technology, the action of smoking can be seen as a technology, as pipes evolved over time, and upon first contact, the Spanish had no words to describe the action of inhaling smoke - it was often described as 'dry drunkenness'

I'd say that the largest impact they had on the modern world is that the way the Aztec subjugated other Mexican natives sort of laid down the blueprint for Spanish conquest. The Aztecs ruled these other groups and so some of them (like the Tlaxcalans) were quite happy to assist Cortes in his takeover. Plus, having a ruling power already in place meant that it was probably easier for the Spanish to establish their authority than if the Aztec Empire hadn't existed- the Spanish would have had to deal with each indigenous group separately. Since the Maya civilization had collapsed centuries beforehand the Spanish had more trouble subjugating the remaining Mayan people in the Yucatan.

I don't mean to sound insulting or belittling of their accomplishments or culture by saying "The main contribution of the Aztecs was to make it easy for Spain to conquer Mexico". But if you really want to be honest about the ways in which the Aztec really did change the world in ways that still echo today, well, that's probably the big one. This subreddit doesn't like the "what if" game but the entire pattern of Spanish colonization could have gone very differently had Cortes encountered a de-centralized political situation.

A number of answers here have hit upon some nice points: a corpus of enduring mythology; linguistic diffusion; the entwinement of Americans and Europeans forming a new mestizo identity; the food, oh my yes the food; and the fact that modern Mexico is, at its core, basically built upon the Aztec state ("Mexico" literally means "the place of the Mexica").

I'm going to be pessimistic though, and say that an enduring legacy of the Aztecs is one of racism; scientific, colonial, and casual. When Cuauhtemoc surrendered to the Spanish-Natives forces in August of 1521, he was following a well-trod template of accepting tributary and vassal status to a conqueror -- a state from which the Mexica had risen before. Yet, that surrender had led to another template, which is how indigenous people -- particularly in the America, but elsewhere as well -- have been, and are still, discussed within the framework of history and popular culture.

The weight of mythology behind the defeat of the Mexica (not the "Aztecs," as 1/3 of the Triple Alliance in fact joined with the Spanish and Tlaxcalans) has so impressed itself into popular understandings of the period that it can be hard to sort out what is known and what is simply assumed. Certain tropes emerge, though so often tied in with the following centuries and the histories of other people of the Americas cast back upon the events of 1519-21. Particularly prominent are the ideas that the defeat was the result of Spanish technological superiority or military doctrine, or that it was the result of the superstitions of the Mesoamericans; or that it was somehow the inevitable consequence of "superior" civilization meeting a "primitive" people.

If that sounds overly dramatic, keep in mind the long pedigree these ideas have. This is Friar Juan de Sepúlveda in his 1547 Concerning the Just Cause of the War Against the Indians:

The Spanish have a perfect right to rule these barbarians of the New World and the adjacent islands, who in prudence, skill, virtues, and humanity are as inferior to the Spanish as children to adults, or women to men; for there exists between the two as great a difference as between savage and cruel races and the most merciful, between the most intemperate [lacking in self-control] and the moderate and temperate, and, I might even say, between apes and men.

"But Sepúlveda was a well-established asshole," you might say, and you'd be right. His sentiments would continue to echo through the centuries though, the idea of the inherent inferiority of Americans growing ever louder. In the 18th Century Buffon and De Pauw would champion the "degeneracy" hypothesis of the Americas, wherein the climate and conditions of the continents enervated the population, leaving the men impotent both sexually and mentally, and the women subservient and strangely fertile. As such, the history of Mesoamerica collected by the early Spanish chroniclers could be dismissed as flights of fantasy. An even more definitive rejection of the possibility of a "civilized" American comes from Robertson's 1777 History of America. While he saw Mesoamerican and Andean people as superior to the other groups in the Americas, he also wrote:

In their highest state of improvement the power of [Mexicans and Peruvians] was so limited, and their operations [of their minds] so feeble, that they can hardly be considered as having advanced beyond the infancy of civil life.

A note in the first volume even provides what must be the most achingly unironic examples of damning with faint praise:

The incapacity of the Americans is so remarkable that negroes from all the different provinces of Africa are observed to be more capable of improving by instruction.

These were simply the high-brow justifications for the dominance of Europeans in the Americas, cloaked now in terms of natural science and mental faculties instead of divine providence. Ironically, the earliest of accounts -- those of conquistadors and monks -- place the various Mesoamericans on the most equal footing in dignity and intelligence. Yet, those could be discounted (when not actively suppressed or lost for centuries) as a result of the credulousness of those chroniclers.

This may seem like nit-picking a bunch of 18th century armchair historians and proto-anthropologists, but the kind of justifications they put forth for the history of the Americas following the collision between the "Old" and "New" worlds has resonated. It is not so hard to look into popular or general history books from the recent past, or even today, which ascribe, perhaps not so much to the "degeneracy" of Americans, but to implicit ideas of passivity and inferiority. That resonance leads to explanations such as the defeat of the Aztec civilization as a result of, as Soustelle put it in 1964, "its material inadequacy or the rigidity of its mind."

The defeat of the Mexica in 1521 created a political vacuum that was filled by the Spanish, but it also greater a greater vacuum, one of explanations for an outcome which -- even today -- seems miraculous and ordained. Digging into the history would show the providence to be not so much divine as mired in banal details and circumstances, but that's not particularly sexy or catchy. Instead we have a convenient package of a small group of intrepid and rational Europeans toppling an empire that was as decadent and irrational as it was primitive. It's a template that would be back-applied to any number of conflicts -- international and personal -- in the seceding centuries. What happened in Mexico in 1521 birthed an ur-narrative justifying not only colonialism, but forming a foundation for a pervasively ethnocentric and perversely whiggish interpretation of history, where a particular society cannot simply exist, but must have an impact that needs to be asserted.

Some sources for further reading:

• Cañizares-Esguerra 2001How to Write the History of the New World: Histories, Epistemologies, and Identities in the Eighteenth-century Atlantic World

• Restall 2003 Seven Myths of the Spanish Conquest

Interesting