This question is quite broad, can you give a time period you're interested in? Do you mean the pre-Columbian period, before the introduction of firearms, or after?
Is there a region you're interested in, or specific tribal groups? The combat among Aztecs and Mayans would not necessarily resemble combat among the Algonquins or Seminoles.
Hi there folks,
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This is my standard note that "Native American" encompasses two continental landmasses with at least 12,000 years of human habitation. Asking about a general, blanket answer about any of the groups which inhabited the thousands of kilometers and thousands of years of the American landmass is a fool's errand.
Disclaimer out of the way and the premise of your question undermined, let's examine one particular place at one particular period of time: the late Postclassic of highland Mesoamerican, i.e., Ye Olde Aztekke Tymes.
While we do not have contemporaneous sources regarding the Aztec order of battle, we do have some frustratingly vague accounts of clashes between various Mesoamerican peoples and the various Spanish entradas. These accounts, unfortunately, almost all stem from the Spanish viewpoint, which means we have little insight into the Mesoamerican thinking with regards to battle order.
For instance, we have this particularly unhelpful account from Cortés, who relates that after approaching a village in the area of Tlaxcala he and his Totonac allies were beset upon by
a great number of Indians began to appear, well armed and uttering loud cries, and commenced an attack upon our people, discharging many darts and arrows...
Though Cortés states the battle lasted "he whole day until an hour before sunset," we receive no more information as to the nature or organization of the attack. Terse descriptions being somewhat typical of Cortés' letter back to Spain, let's instead turn to Bernal Díaz de Castillo, who was part of Cortés' company and also took part in at least one other expedition from Cuba to the American mainland.
In his accounts of the initial clashes of the Spanish and their Totonac allies with the Tlaxcalans, an immediate difference is noted. First, he describes multiple battles over the course of several days with variously sized groups of combatants, rather than the sparsely described couple of clashes with "more than one hundred thousand warriors" that Cortés claims before quickly sequing to the the Tlaxcalans suing for peace and somehow immediately shifting their allegiances to the king of Spain.
In contrast, Díaz del Castillo recounts an initial clash with a vanguard of 6000 Tlaxcalans who:
set up a most terrific noise with their drums and trumpets, and yelled awfully. They then let fly their arrows, threw their lances at us, and upon the whole were most daringly valiant.
This was followed by a large engagement after the initial clash was repulsed with gunfire and the Tlaxcalans withdrew beyond harquebus range. In the subsequent larger engagement, Díaz del Castillo says that some 40,000 Tlaxcalans "plied their bows and lances with great dexterity" and states that the Spanish/Totonac forces endeavored to maintain a defensively closed position, only break it to rescue on Spaniard whose horse was cut down beneath him. This is actually the famous account of a horse being near decapitated by a macuahuitl blow, and Díaz del Castillo makes a remark that the Tlaxcalans seemed intent on capturing a horse dead or alive during the battle. The capture of a horse makes sense both from a basic military intelligence sense, but also from a symbolic approach, as Díaz del Castillo reports that parts of the dead horse were sent to various towns as trophies. Horses being killed, dismembered, and sent to various towns as a symbol not just of the vulnerabilities of the Spanish, but also as a way of de-mysticizing these strange new creatures was later seen in conflicts between the Spanish/Tlaxcalans and the Aztecs.
What do these early clashes with the Tlaxcalans tell us about battle order among late Postclassic highland Mesoamerican groups? Well, not much, because the Spanish conquistadors were not particularly astute in their recollections of the nuances of their engagements with Mesoamerican forces. Instead what we get is fairly general descriptions of an initial flurry of arrows, sling stones, and atlatl darts followed by a increasingly vague descriptions of a fierce melee, with perhaps one or two particular incidents noted. Yet even those notable clashes are presented more as summary report of what made them notable (often a mounted Spaniard extended to far forward and getting surrounded) followed by an equally terse accounting of what the Spanish did in response (e.g., we moved forward and rescued the person).
For instance, when Díaz del Castillo recounts the defeat of the Spanish on the causeway wherein Cortés was pulled form his horse, we likewise have not much detail other than, again, an initial missile volley (this time including attacks from canoes upon the causeway), which was then followed by a crushing attack forcing the Spanish into a defensive retreat. Díaz del Castillo does precisely say that sixty Spanish were captured, many more wounded, and implies Cortés was dragged off his horse, but neglects details as to how, and is equally vague in his rescue, basically just stating that the Spanish rushed forward, secured him, and threw him on the back of a horse to be carried back to safety.
Missing from all of this, of course, is any remarks as to how the Mesoamerican allies fighting alongside the Spanish were arranging themselves, but this is typical of the Spanish accounts. The conquistadors spare no expense on the page to recount in vague but glorious terms the bravery and valor of their compatriots, and their battlefield discipline, but their indigenous allies are all too often invisible. The uncharitable and uninformed interpretation of the Spanish silence on this topic would be to say the native allies were not significant, but this is neither realistic nor consistent with the accounts. The more realistic approach is assuming the Tlaxcalan and other Mesoamerican allies of the Spanish continued to operate alongside, but still independent of the Spanish forces. Simply put, it is not realistic to assume the thousands of indigenous allies of the Spanish were efficiently commanded by the Spanish. The absence of these troops from the Spanish narratives instead reinforces the notion of these forces operating with little to no input or understanding of their tactics from the Spanish corps.
Even when we have an early Nahuatl account of battle, it does not give us much understanding of the order of battle, instead mirroring the Spanish accounts in recording singular heroic deeds and generalized descriptions. So we have the glorious mention of a Tlatelocan who was lanced by Spaniard, but grabbed hold of the spear, leading to he and others unhorsing his attacker and slaying him. We have the mention of another Tlatelolcan named Tzilacatzin who rallied an offense, with he and others slinging stones to slay several attackers, then continuing to fight so fiercely on other occasions that he wore disguises so as not to easily singled out.
Both of those accounts are from the Florentine Codex which was compiled a couple decades after the battle for Tenochtitlan, and which (as may be evident from above) contains a significant bias against the Mexica of Tenochtitlan, and a strong bias towards the Mexica of Tlatelolco, the sister city that the former conquered a couple generations earlier. So there's an element of settling scores by discounting the Tenochca and favoring the Tlateloca.
While waiting for some of my colleagues to dive in, you might be interested in this previous thread about large scale Native American conflicts prior to contact.
As others have pointed out, "Native American" is entirely too broad. It's important to keep in mind that the Native American people were not a monolithic force, and in fact, were made up of many different cultures. Weapons and tools were built using what was available to them, which included animal bones, stones, shells, etc. So the weapons and tools of tribes from Florida would look quite different from those made by those in the Carolinas.
When Hernando de Soto and his men arrived at Florida, they were both horrified and impressed by the quality of their weapons and fighting skill of the Natives. They described the Natives as “exceedingly ready with their weapons” “warlike and nimble” and fearless.^1 He goes on to state that the Natives easily avoided their arrows, crossbow bolts, and arquebuse rounds, and how their weapons destroyed the armor of the Spaniards. ^2
When colonization efforts began in earnest, local tribes would ally themselves with the colonists and would even join their armies. Friar Francisco Alonso de Jesus when writing to the King of Spain noted that the Natives, while unskilled in the techniques of agriculture, manufacturing, and other skills common to the Spanish, they were very intelligent people that learned quickly.^3 Therefore the Spanish "hired" the Natives to build structures for them, clear roads, and fight in their battles. I put hired in quotes because while Christian Indians were required to be paid as subjects of the crown, they were often paid in small trinkets, a single meal for an entire days work, or wages low enough to only pay for a single pastry.^4
So I guess to answer the question, no, they generally were not just chaotic or savage. Again, cultures, tools, weapons, etc, could vary greatly between communities, but from the accounts we have from those that encountered Native Americans, they knew their way around a fight. Many of the tribes were constantly fighting among themselves. Steven J. Oatis in his book, A Colonial Complex: South Carolina's Frontiers in the Era of the Yamasee War, 1680-1730, notes that the Natives saw war very differently than the Colonists. They didn't see it as a conflict with a beginning and end or an overall goal. It was a way of life for them.
As /u/mizzamir already pointed out, "Native American" is fairly broad - even in the most narrow definition of applying to what would later be the United States - so it is hard to give any answer which is representative of the 'whole', but while others will hopefully be able to weigh in on specific regions, I would recommend that you check out our FAQ Section on 'Pre-Columbian Warfare' which has several relevant answers from earlier threads covering some of the areas!