Yes, there was an elite class in Aztec society. I'll let Fredrick Hicks, who wrote extensively on class in Late Postclassic Mesoamerica, describe the basic arrangement:
The ruling class was drawn from (but was not coterminous with) the hereditary nobility, which had the right to receive land and the tribute and services of peasants settled on the land. These peasants were not bound to the land; they were provided with subsistence plots, but could abandon them if they found somewhere else to go. The nobility was grouped into lineages, which often had widely dispersed branches, but the commoners were not, and there were no recognized kinship ties linking nobles and commoners. Within the nobility, certain lineages were royal lineages, and from them the rulers of states were drawn. The head of the royal lineage held the rank of tlahtoani, or king, but it was actually the title he bore, not his rank, that made him the ruler. All noble lineages were internally ranked; heads of noble houses held the rank of teuctli, and lesser nobles were pilli.
Although this was from the 1996 text, Ideology and the Formation of Early States, Hicks was an old school Mesoamericanist (he passed away in 2012 and did much of his formative work in the 1970s), hence the reliance on direct comparisons to European nobility, and adoption of such terminology. But he's also not wrong, so let's unpack this explanation a bit.
First, we need to acknowledge that the above passage from Hicks is describing the state of Aztec class and state in the early 16th century, right before the tumultuous reshuffling of power arrangements in the colonial era. While class relations in the Aztec dominion had been at their most solidified at this point, the history of the Aztecs is one of incredibly fluidity and cultural change. The Mexica went from vagabonds to refugees to subject polity to the overwhelmingly dominant force in Mesoamerica in the course of about 200 years. We see this most keenly in the presence of cuauhpilli (lit. Eagle Nobles), in the early Aztec period. These were men who, through their prowess on the battlefield, could be elevated to a sort of "life peer." They enjoyed the benefits of nobility, but their status was not hereditary (though this would not prevent forming marriage alliances with an existing noble family). By the time of Motecuhzoma Ilhuicamina (1440-1469), however, this practice had faded out, and by the time of Moteuchzoma Xocoyotl (1502-1520), it was a distant memory.
Second, we do need to explore the difference between tetecuhtin and the pipiltin (Nahua plurals are... not like English plurals). As per above, one way is to think of them "upper" and "lower" nobility, but this was not an innate, de jure distinction. Another lens though which to seem them is in closeness to the currently ruling dynasty. As the upper echelons of Aztec society most often formed political alliances through dynastic unions (i.e., marriages), what we see is a snowballing effect of powerful dynasties, like that of the Tenochca-Mexica, growing larger and more diffuse over time. There then becomes a sort of chicken-or-egg question as to whether a teuctli was such because of their direct connection to a powerful ruling dynasty, or whether becoming a powerful ruling dynasty made its members teteuchtin.
Perhaps a better way to look at Aztec nobility is through a lens that Hassig (1994) uses in Mexico and the Spanish Conquest. Writing of the attempt by the Spanish to set up a young son of Motecuhzoma Xocoyotl as his successor following the ruler's death, Hassig notes a fundamental disconnect between Spanish and Aztec concepts of nobility at the time. While European elites saw their nobility as innate and in their blood, Aztec society saw nobility as a performative role. To claim nobility, particularly high nobility, one must act as a noble. This meant success in war, which obvious a child could not accomplish, but it also meant being able to represent a greater concept than just your own lineage. A pilli may be of an elite dynasty, but a teuctli represented an entire people. It is not a coincidence that, upon the formation of the Aztec Triple Alliance, the three rulers of the consituent polities took titles claiming dominion over broad and nebulous swathes of peoples: the Culhuas, the Chichimecs, and the Tepanecs (the last more concrete, but Tlacopan was always the junior partner).
How did this all work on practical level? Well, in one sense it comes down to land ownership. The common pattern in Central Mexico was to have lands held communally, but with individual plots granted to families to do with as they pleased so long as they continued to maintain the land. This was fundamental to the calpulli system which made up not only the urban areas, but also rural settlements, and participation in this corporate land-holding system was how the macehualtin who made up the bulk of the population lived. Not so with the nobility, who -- if they were defined by anything -- were defined by the fact of holding private lands.
In the classic sense of the rich getting richer, working private lands with mayeque (tenant farmers) and tlacohtin (slaves) could be used to produce luxury crops, even as elite status allowed nobles to demand tribute, which could range from basic foodstuffs to luxury goods. To return to the aspect of Aztec nobility being performative, the easiest way to acquire and expand these privately held lands was through warfare. Success on the battlefield brought not only social recognition, but also opportunities for seized land (and people) to be granted to the victor. This was particularly true in the early Aztec Imperial period, when conquests were close at home, whereas later conquests were more about extracting high-value tribute.
So yes, there was an Aztec nobility. It was built, on a practical level, through the demonstration of force which led to wealth. The elite class, however, was also beholden to cultural notions of behavior and the burden of representing both a lineage and a people. Sahagun's General History of the Things of New Spain has an entire chapter devoted to laying out the ideals of what it means to be an "honored noble," as contrasted with a "bad noble." The virtues extolled include honesty, modesty, eloquence, and compassion. A good noble would be obedient to his betters and he would inspire others though the moral force of his character. This was the complicated nature of Aztec nobility, to be a warrior, landlord, student, teacher, ascetic, and luxurious all at the same time.
This was a bit sprawling, so let me know if you have any follow-up questions. Here's some sources to peruse not already mentioned:
Evans ST (1985) The Cerro Gordo Site: A Rural Settlement of the Aztec Period in the Basin of Mexico. Journal of Field Archaeology 12(1)
Hicks F (1974) Dependent labor in prehispanic Mexico. Estudios de cultura náhuatl 11
Hicks F (2012) Governing Smaller Communities in Aztec Mexico. Ancient Mesoamerica 23(1)
Rounds J (1977) The Role of the Tecuhtli in Ancient Aztec Society. Ethnohistory 24(4)