Answering how the Aztecs celebrated the New Year necessarily means we need to identify which day was the Aztec New Year and what, specific, festivities were held that day. The problem is that there does not appear to be a one-to-one mapping of what Western cultures would consider New Years Eve (with its attendant celebrations) and Aztec calendric practices. Postclassic Mesoamerica absolutely had regular festivals and rituals at specific times on their calender(s). Mesoamerican calendar also absolutely had a point at which the cyclical count of days repeated anew (many of them, infact), but understand this question we'll need to break down how time working among the Aztecs.

Just to start, let's establish which calendar we are talking about, since Mesoamerica used two. The first, a 260 day rotation was known to the Aztecs as the tonalpohualli (20 named months of 13 numbered days) and is often referred to as a "ritual" calendar, since it featured very often in divinatory and oracular practices. We are not talking about the toanlpohualli. The other calendar system was a 365 day rotation, the xiuhpohualli* (18 numbered months of 20 named days), which should be more familiar, seeing as how it was a "solar" calendar matching the rotation of the Earth around the sun. If the tonalpohaulli was the calendar of priests and magic, the xiuhpohualli was the calendar of farmers and agriculture (though it also had enormous ritual significance, as we shall see). If we are to ask about a "new year" then we are asking about when the xiuhpohualli ended one cycle and began anew. Simple, right?

No.

There are some basic problems when dealing with dates and calendars in Mesomarica. The first is that there are several different correlations for translating Aztec calendar dates into Julian and Gregorian calendar dates. The most famous is from Alfonso Caso, who took the surrender of Cuauhtemoc to Cortés as a fixed date from which to count back. The Julian calendar date was 13 August 1521, and we have an Aztec date of the day 1 Coatl in the year 3 Calli. From there, Caso was able to establish a fixed correlation between the Julian Calendar and the Aztec Calendar, and thus we can say that, per the Caso correlation, that the "month" of Atlcahualo, which is normally seen as the first month of the xiuhpohualli, started on February 14.

Unfortunately, two early an important sources give different dates for when Atlcahualo started. Sahagún states February 2 was the start date, but Durán, writing just a few decades later, says the date was March 1. Sahagún is earlier and avoids some of the ethnocentric baggage Durán ladles on, so I'm more inclined to believe the former over the latter, but that still leaves the problem of neither date matching up with Caso's correlation.

This brings us to our second problem, which is that different polities in Central Mexico probably used different calendars; there was no one single overarching, universally agreed upon system, but instead variations on a generally agreed upon basic premise. There are also factors to take into account like poor record-keeping after the Conquest and whether or not leap years were used, but variants in the calendar system used by even closely aligned groups is probably most relevant to your question. We can see the diversity of practices even from the ambiguity from how Sahagún waffles on which god was worshiped during Atlcahualo, as well as the some nomenclature issues:

The first month of the year was called, among the Mexicans, Atl caualo, and in other parts Quauitl eua. This month began upon the second day of February, when we celebrate the Purification of our Lady. On the first day of this month, they celebrated a feast in honor -- according to some -- of the Tlalocl gods, whom they held to be gods of rain; or -- according to others -- of their sister, the goddess of water, Chalchiuhtli icue; or -- according to still others -- in honor of the great priest or god of the winds, Quetzalcoatl.

That is some top level ambiguity, but at least we can be assured that Atlcahualo was the first month, right?

No.

Or at least no fully, and for this we need to talk about Year Bearers. Up above I wrote that the surrender of Cuauhtemoc occured in the year 3 Calli, but what does that mean? Well, each year was named after the xiuhpohualli day it started on. Because of the way the mathematics of having a 365 day calendar divided by 20 day names plays out means that only every fifth day-name could start the year. As a result we only have years named after the days signs Calli (House), Tochtli (Rabbit), Acatl (Reed), Tecpatl (Flint Knife), proceeding in a sequence like 1 Calli, 2 Tochtli, 3 Reed, 4 Tecpatl, 5 House, etc. So the first day of the year, the year bearer, is the first day of Atlcahualo, right?

No. Well, maybe.

There's some ambiguity about when days were actually considered to start in Mesoamerica (at noon, dawn, midnight, etc.) which complicate things, but the main thing here is that setting Atlcahaulo as the first month of the year entails some (not a lot, but some) amount of debate. Caso, he of the famous correlation, argued for making the month of Izcalli, usually placed as the last month, as the first month of the year. Despite the fact that early sources almost universally place Atlcahualo as the month that kicks off another cycle of the xiupohualli, there's still some disagreement on this, and it comes down to the year bearer and whether it should be the first day of first month, or the last day of the last month. If the former, then Atlcahualo is undeniably the first month of the year, if the latter, though, should Izcalli actually be seen as the first month of the year, since it contains the year bearer?

Complicating all of this is that between Izcalli and Atlcahualo are the nemontemi, the five "empty" days of the Aztec calendar. You might have noticed that a solar year comprised of 18 months of 20 days adds up to 360, not 365. In addition to the 18 months of the xiuhpohualli there were also five days added at the end (or start?) of the calendar. This does argue for Atlcahualo being the start of the year, but does not resolve the question of which day was the year bearer (and thus the approximate to our New Years Eve/Day). Also, some sources cite Tlacaxipehualitzli, the month after Atcahualo, as the first month, but let's just elide right on over that.

Since we are not going to settle decades of academic wrangling here, let's focus more on the heart of your question: what did the Aztecs do around the new year (defined as when the xiuhpohualli started a new cycle, which may or may not have been at the end of the month of Izcalli or the start of Atcahualo)? That's an easier question to answer, given that our early sources were ready and able to talk about what went on during those months. Let's start with what went down during Atlcahualo.

**Answering how the Aztecs celebrated the New Year necessarily means we need to identify which day was the Aztec New Year and what, specific, festivities were held that day. The problem is that there does not appear to be a one-to-one mapping of what Western cultures would consider New Years Eve (with its attendant celebrations) and Aztec calendric practices. Postclassic Mesoamerica absolutely had regular festivals and rituals at specific times on their calender(s). Mesoamerican calendar also absolutely had a point at which the cyclical count of days repeated anew (many of them, infact), but understand this question we'll need to break down how time working among the Aztecs.

Just to start, let's establish which calendar we are talking about, since Mesoamerica used two. The first, a 260 day rotation was known to the Aztecs as the tonalpohualli (20 named months of 13 numbered days) and is often referred to as a "ritual" calendar, since it featured very often in divinatory and oracular practices. We are not talking about the toanlpohualli. The other calendar system was a 365 day rotation, the xiuhpohualli* (18 numbered months of 20 named days), which should be more familiar, seeing as how it was a "solar" calendar matching the rotation of the Earth around the sun. If the tonalpohaulli was the calendar of priests and magic, the xiuhpohualli was the calendar of farmers and agriculture (though it also had enormous ritual significance, as we shall see). If we are to ask about a "new year" then we are asking about when the xiuhpohualli ended one cycle and began anew. Simple, right?

No.